In Fullmetal Alchemist Brotherhood, we saw that the core of a homunculus is a philosopher's stone, but ironically, they still can't perform any alchemy.
Why is it like that?
In Brotherhood (this does not totally hold true for the 2003 anime, so the reason is somewhat different), in order to perform alchemy, one must have a personal Gate of Truth. This is shown to be evident at the end of the series, when:
Edward Elric gives up his Gate of Truth in exchange for Alphonse, and can no longer perform alchemy as a result of losing it.
However, manufactured homunculi (i.e. not Wrath or the reborn Greed✝✝) are made up of many souls, since:
The Philosopher's Stone that composes them is powered by thousands of trapped, living souls. This power is what gives the homunculi their life energy as well as permits use of their regenerative (and other) abilities.
These many souls (generally, there are some exceptions) cannot think or act on their own, and thus the homunculus does not have his own Gate of Truth. This being the case, they cannot perform alchemy.
✝✝ It is possible that Wrath and the reborn Greed, having been born human and later becoming homunculi, could perform alchemy if they sought to do so. However, neither of them have any desire to learn it, and there is no official word one way or another.
The reason why Homunculi cannot perform alchemy is because they do not have their own individual soul. They are all a part of Father's soul, they are his 7 deadly sins (Wrath, Envy, Lust, Greed, Gluttony, Sloth, and Pride) and are a part of him. Because only human souls have gates (even if they are not alchemists) and Homunculi do not have their own souls, they cannot perform alchemy. However, at the end of FMA:B:
We see Father at the gate in his Dwarf form. This proves that Father is the only true homunculus that is actually a human.