Having read a few discussions regarding the Plantagenets and the ascending line of the monarchy since, I became curious about whether another group/family had a 'more' legitimate claim to the throne (asked and answered, though not sufficiently in my opinion) or if another family could become the monarch. I don't want to base a question on an assumption, but I am curious all the same if by whatever manner the Windsors could be replaced - for lack of a more appropriate term.
Note: I don't think that what I'm going to offer is a particularly good answer, but I think it is a worse comment.
I think the key to the answer is the Glorious Revolution in which England expelled James II and invited William and Mary to rule England.
The legal basis for the crown is the 1701 Act of Settlement, which I missed, but @Spencer caught - hat tip to @Spencer
Aside: @Oldcat's comments on the Young Pretender and the Old Pretender are very pertinent. I approach the question through the lens of an American fascinated with British Constitutional evolution in the period. I think that @Oldcat's comment is probably closer to the OP's actual intent. For me, the British Constitution prevents any pretender from having a legitimate claim on the throne.
Although the Glorious Revolution is vital to understanding Great Britain, in my opinion it isn't the actual event of the Revolution, but the events of the following years that really define the British Constitutional Government. I'd recommend starting with Edmund Burke Who is brilliant, but not easily quoted or summarized. With respect to the current question, I think that Reflections on the Revolution in France is vital, and I believe that Thoughts on the Causes of the Current Discontents may also be useful. I actually haven't looked at this question in a couple of years, but I believe that the (British) Bill of Rights of 1689 is probably important. Essentially Burke argues that the English invited William and Mary to rule only after James II had invalidated his rule, and that the English agreed to be bound to the rule of William and Mary. He is quite adamant that unlike the French, the English Constitution is not based on a social contract, and that the English don't have the privilege of selecting Monarchs willy nilly.
Updated aside: I haven't researched this period for about five years. I remember finding a paragraph from Burke that I thought summarized the question admirably (once I decoded Burke's prose). I'm truly frustrated that I cannot find that quote anymore (I've begun to wonder if it was not Burke, but someone else, but I cannot remember).
″Our political system is placed in a just correspondence and symmetry with the order of the world, and with the mode of existence decreed to a permanent body composed of transitory parts; wherein, by the disposition of a stupendous wisdom, moulding together the great mysterious incorporation of the human race, the whole, at one time, is never old, or middle-aged, or young, but in a condition of unchangeable constancy, moves on through the varied tenour of perpetual decay, fall, renovation, and progression. Thus, by preserving the method of nature in the conduct of the state, in what we improve we are never wholly new; in what we retain we are never wholly obsolete.″ Reflections on the Revolution
If you're looking for a good overview, I would suggest Mike Duncan's Revolutions podcast and website - he covers the English Civil war fairly well. Although the Civil War is distinct from the Glorious Revolution, I understand the Glorious Revolution better for understanding the Civil War.
I also recommend @Kobunite's answer summarizing the effect of the Bill of Right. I think Kobunite's summary of the various clauses is clear, helpful and accurate. I wish I had the skill and the knowledge to explain why it is difficult for a modern reader with a modern reader's understanding of constitutional theory to read the Bill of Right and understand the assertions. Modern readers rely on assumptions and concepts that were formed as a result of the Bill of Right; the contemporaries didn't have that advantage. Quite frankly the analytical summary that Kobunite provides is (a) why I find the period fascinating and (b) part of what makes this period so vital to the understanding of the American government.
Today the British monarchy is an almost entirely ceremonial position. All the real political power in the country is invested in Parliament (mostly the House of Commons) and the court system. The Queen is left with only the ceremonial "Head of State" duties that in the US are performed by the President (or more often in fact delegated to the First Lady and/or the Vice President). Greeting foreign dignitaries, attending state funerals, etc.
For this reason, frankly there's no real incentive for anyone to try to change things in the violent manner that would typically be implied in an "overthrow".
Parliament can however, and in fact often does, pass laws changing both the personal requirements for a monarch, and the succession requirements. This generally has to be coordinated with the other dominions to avoid inconsistencies. This was most recently done in 2011 to allow for a fully gender-neutral succession. It would similarly be in their power to depose an unacceptable monarch (as was effectively done in 1936).
The most likely scenario for any kind of radical change though (and probably still not very likely) would be an abolition of the monarchy altogether, with the Head of State position becoming an elected one (or perhaps selected by Parliament). This gets discussed a fair bit in England particularly whenever any personal problems of Royal Family members make news, but support for that idea recently seems to bump around between only 15%-25%.
The current monarchy in the United Kingdom is very attuned to the public relations between that monarchy and the people of England (yes, and I mean only England). Other public relations behaviour is far more muted in comparison. This responsiveness emerges from a number of public relations disasters, including the monarchy's German heritage which caused the formation of the House of Windsor itself (https://en.wikipedia.org/wiki/House_of_Windsor#Foundation).
This sensitivity to the public is probably due to 20th century incidences of dethronement in Europe after mass public unrest, generally leading to a parliamentary republic, occasionally and briefly leading to systems of workers councils. This, at least, is the animating threat that lies behind the the House of Windsor's activity, service to the nation, etc. I would suggest that fear from the Windsors indicates at least one viable method of removal, within or without the constitution: mass public outrage which either parliament attends to, or which also destroys parliament.
To belatedly expand, Graves and Hodge’s The Long Week-End: a social history of Great Britain 1918-1939 not only notes the general strike, but also that the British republican flag was displayed in 1935 during Edward’s unwillingness to conform.
I can envisage a perfectly valid scenario for the abolition of the British monarchy - although an unlikely one.
A radical republican government is returned to power with an overwhelming majority. It passes a Bill to abolish the monarchy and replace it with an elected President. Now, it is not law until it has received the Royal Assent, and the sovereign is not bound to grant it, merely expected to.
If the sovereign refuses her/his assent, the Government resigns, triggering a General Election. The result is the same - a huge majority for the republicans. In theory the monarch could ask anyone in Parliament to form a Government, but this would almost certainly not work, given the size of the (now) Opposition). No legislation would be passed, a vote of No Confidence in the Government would be passed, triggering yet another General Election. The British are generally slow to "man the barricades", but even they would probably be driven to it by then.
It is, therefore, far more likely that the sovereign would accept the will of the people, expressed in Parliament, and sign the Abolition of the Monarchy Bill into law, thus abolishing her/himself.
With British pragmatism, all this would be accompanied by (probably secret!) discussions as to the terms of the royals' departure - financial arrangements, leave to remain in/return to the UK, agreements preventing any "Pretenders" later challenging the law etc. It is an unlikely, but perfectly possible scenario.
The only main difficulty I see is in the beliefs and views of the current Queen. She is devout, and genuinely believes herself called by God to fulfil her role, for life. She might regard agreeing to the abolition as a breaking of her Coronation Oath. On the other hand, she is reputed to have said in the past that they "would go quietly", if asked.
So yes, it would be constitutionally possible to abolish the Monarchy. It's just not going to happen, at least any time soon.
Well, the monarch holds that position at the pleasure of Parliament. The constitutional way to replace the monarch is to persuade Parliament, and a sufficient fraction of the population, that installing you as monarch would be to the national advantage. The limited role of the monarchy, and its long tradition and popularity make this impractical at present.
Even if you could persuade P&tP that the monarch should go, convincing them that you'd be a better idea than the abolition of the monarchy would be very hard. The British are quite aware that the monarchy is an anachronism and somewhat silly, but they currently prefer it to the alternatives, largely because of the perceived qualities of Elizabeth II. When she dies, there will be an upsurge in republicanism, and while I'd expect Charles to succeed, it isn't a given.
Seizing the crown by small-scale force would be futile. P&tP would take no notice of you, and would rally to whoever was the surviving heir. There are at least 5,000 descendants of Sophia, Electress of Hannover, so there would be an heir, although sorting out who it was might be tricky. In any case, you'd be arrested or killed by the military.
If you were able to start a full-scale revolution (difficult, at best) or mount an invasion, and by one of these means overthrow the whole governmental structure, you could establish a new one with you as monarch, but hanging onto the position could be hard work. We did revolution in the 17th century, and we didn't like the results.
If only you want to acquire the use of the monarch's powers, that's more practical. One way is to marry someone who is in line for the throne, and allow time and a few staged accidents to do their work. Your best bet would have been to marry Prince Harry, and arrange for an accident for his elder brother and his descendants. As time has passed, there aren't any good candidates for this stunt at present. There is no way to acquire the throne for yourself via marriage.
However, the easier way to acquire the use of the monarch's powers is to become Prime Minister. This is completely legal, doesn't require violence, and can be accomplished in about 20-30 years of hard work in the bowels of British politics.
The law of succession has always been open to some interpretation. See this mess here: https://en.m.wikipedia.org/wiki/Alternative_successions_of_the_English_and_British_crown. Making up some codswallop about how your guy was the 'real' king was what you did if you didn't like the current one.
In principle, no law can be passed without royal assent. The convention is that it's waved through without the monarch even being physically present. However conventions are not enforceable in the courts, so if a monarch really didn't want to sign a law, who knows?
The closest Britain has come in recent times is Edward VIII. His government, for several reasons, threatened to resign unless he abdicated. He folded. If he hadn't, he might have faced a revolution, but legally there would have been no recourse whatsoever.