Could a percussion rifle such as the springfield m1861 be made somewhere between the 15th-17th century? If so how would it be made?
A percussion rifle needs two things - a percussion cap and a rifled barrel.
Percussion cap depends on the discovery of fulminates to work. It happened in 1800.
Rifled barrels appeared significantly earlier, but were the pain both to produce in significant amount and to load in muzzleloaders.
Of those two, the discovery of fulminates seems to be a more limiting factor. Unless something changed significantly in some alternative world, I do not see percussion firearms managing to appear earlier then they did - in limited numbers in 1820s and tried out by the militaries in 1830s (there existed a percussion version of M1819 Hall rifle).
If this is a time traveler scenario, I think that all mechanisms after the wheellock are of comparable mechanical complexity to percussion mechanism. So a time traveler would be able to have one custom rifle made with a percussion mechanism any time after 1500s.
If he has means to produce mercury fulminate primers (which is its own big assumption) and wants to arm a significant amount of people with percussion firearms, he is better to go with percussion muskets - he will not be able to have standardized rifled barrels of a single caliber made any time before 1800s.
discovered in 1799 by the British chemist Howard (sic)
The answer to the question comes from a later developement:
However a system which was subsequently developed as an alternative - the Maynard tape primer uses a roll of paper (sheltered inside the mechanism) with primer enclosed in "pimples", which are fed by a ratchet mechanism to the hammer plate when the hammer is cocked. When the trigger releases the hammer, it causes the pimple to violently deflagrate - the subsequent explosion is then fed through a channel igniting the powder in the barrel (and trimming the tape at the same time - neat).
Whilst the modern primer (Armstrong's mixture) would require advanced chemistry not available in the 15th century, black powder can be set off by percussion. In principle there is no reason that a similar system could not have been invented as early as metalurgy (with precision smithing and the ability to temper springs), gunpowder and paper existed in your world.
There is an exciting, although somewhat impractical means of making primers using the technology of the times: fulminating gold!
Fulminating gold was the first high explosive known to man and was first noted in western alchemy as early as 1585