The reference is to the Book of Common Prayer, the standard Anglican liturgical book. It was a locus of many doctrinal squabbles over church teachings and practices. The edition in use in Macaulay's time (mid 19th century) would have been the 1662 version with some minor additions. In the era, attempts to revise the prayer book were stalled due to the conflict between low-church evangelical types and high-church Anglo-Catholic types, so the version actually in use was not up to date with anyone's thinking.
One of the Restoration-era additions has the following title:
A form of prayer with fasting, to be used yearly upon the Thirtieth Day of January, being the Day of the Martyrdom of the Blessed King Charles the First; to implore the Mercy of God, that neither the Guilt of that sacred and innocent Blood, nor those other sins, by which God was provoked to deliver up both us and our King into the hands of cruel and unreasonable men, may at any time hereafter be visited upon us, or our posterity.
Charles I had been executed in 1649 after a civil war, with his son Charles II regaining the throne in 1660. In between, the kingdoms were governed as a commonwealth with a Lord Protector instead of a king. An important axis of the situation was that both Charleses favoured Catholicism and an expansive role for the king himself, whereas the other side had a Puritan/Presbyterian flavour and were willing to cut kings' heads off. This set of prayers, inserted under Charles II, reflects a romantic tradition of Charles I as an unjustly slain hero whose death went against the divinely ordained office of kingship.
In the prayers, Charles is presented as the Lord's anointed and a martyr, through various scriptural quotations and pleas for divine mercy, such as that the land "may be freed from the vengeance of his righteous blood". Much of the language used is more commonly seen applied to Jesus himself, and the several paeans to Charles's virtues were surely ridiculous in the ears of a Whiggish historian like Macaulay. Macaulay's book shows an effort to take a more realistic view of Charles's strengths and weaknesses, an overall tone favouring a legal limited monarchy, and a distaste for royal interference with religious observance.
Many versions of the BCP are available online or in libraries. Some online examples include a modern reproduction of the 1762 Baskerville edition which includes the 30 January prayers, and an OUP 1839 printing that is closer to Macaulay's time. Many, many more are described and linked at justus.anglican.org.
The "Charles the Martyr" prayers, and two other related additions (for the Gunpowder Plot and for the Restoration), were removed in 1859. This was done by combined authority of a Royal Warrant for revision of the BCP, and an Act of Parliament (22 Vict c.2) repealing various statutory mandates for observances of the days in question.