In Desmond Seward's book on the hundred years war, he introduces Henry thus:
In the national legend Henry V remains the most heroic of English
Kings. He is the glorious conquerer who broke the French chilvary at
Agincourt and won the throne of France for his son's inheritance.
Henry V is obviously best known for his military conquests. His military career began in his teens, fighting successfully against the Welsh rebel Owain Glyndwr and his allies the Percies and Mortimers. As king, he won a famous victory at Agincourt and in later campaigns, so successfully conquered territory that he forced the French into the Treaty of Troyes that stipulated that the French crown should pass to his line upon Charles VI's death.
Henry's personal and political skills also did much to cement his reputation. He had a much better relationship with parliament than did his profligate father and other predecessors. He was able to put aside personal differences, willing to appoint to powerful positions the sons of men who had been executed by his father and winning their loyalty. Indeed, it was Edwin Mortimer, former heir of the deposed Richard II, who informed Henry of the Southampton plot to depose Henry and replace him with Edwin. Also, Juliet Barker wrote that he was particularly scrupulous in ensuring that he did everything possible to ensure that there was divine justification in his military actions (but her Agincourt book does tend to read as a panegyric to Henry).
Henry predeceased Charles, leaving his son Henry VI to inherit the French crown and his brother John as regent to take England to the peak of its power in France before its inevitable fall. Perhaps Henry V's early death helped his reputation, as he lacked the opportunity to suffer the reverses that had afflicted Edward III and the Black Prince after the successes of Crecy and Poitiers early in their careers.
Culture undoubtably played a role in Henry's reputation. As with Richard III, the public's impression of Henry V owes much to the works of Shakespeare, so we are thus left an image of Henry both as heroic conquerer and (unfairly) as dissolute youth. Shakespeare, as usual, borrows from earlier works. Barker wrote that:
As the last vestiges of English power in France were slowly but
inexorably eradicated, people looked back to the glory days of
Agincourt with nostalgia. Ballads, chronicles and plays in English
were written for an increasingly literate bourgeoisie preserved the
memory of the victory and served as a rallying cry for future wars in
The need for a pristine image of Henry V to serve as an English propaganda tool (including as recently as WW2) has led to some of his more questionable acts being glossed over. I'll leave the last word to Seward's final sentence of the introductory paragraph quoted above:
In reality he displayed a number of markedly unheroic qualities and,
in a gentlemanly, medieval sort of way, he had more than a little in
common with Napoleon and even Hitler.