What happened to the bullet that William Stryker hit Wolverine with?

The movie doesn't substantively answer this question. There are multiple possibilities, all consistent with various aspects of Wolverine depicted over the years in the X-men cinematic universe. (We can attempt a proof by reasoning/deduction strategy, but anything which isn't explicitly addressed in the film, can very easily transition into speculative territory.)

• With sufficient velocity of the Adamantium bullet, it would possess sufficient kinetic energy to penetrate through the skull surface on front, pierce through the brain, and escape out from the back surface of the skull. This way, it would make two holes in the skull, but neither would leave any visible sign on the outside, since his skin heals back, for example, as seen here, between 0:10 (where there are two bullet marks on Wolverine's forehead) and at 1:00 (where his forehead is clean, no injury marks). It makes sense to believe the bullet wounds at the back of his head would heal similarly too.

This would be consistent with Wolverine's memory loss, as per possibility 1 of this post:

Memory is stored as a series of chemical changes or connections in the Brain.

When the bullet pierced his brain, it damaged tissue. While his body healed that back, the chemical responses (memories) stored in the tissue were lost indefinitely. So although his brain healed back, and other functions were generated normally - like speaking and walking and so on, the long term memory area suffered significant damage.

This was Styker's objective too, as he declared, while preparing his gun, filled with adamantium bullets:

"His brain may heal, but his memories won't grow back."

• Alternatively, with lesser bullet velocity, the bullet would pierce the skull surface once on the front, but would not have sufficient energy to pierce through the back. In that case, the bullet still continues to be inside his skull, ans while cells and tissues around it would heal, part 2 possibility still exists:

When the bullet pierced his brain, it cut through brain tissue and got lodged somewhere in his brain. So while the other tissues healed back up, body functions and actions were resumed, the part where the bullet was lodged might have been the area where long term memories are stored. So that is the reason his memories might never have reformed.

However, this possibility does have one weird consequence. There have been various instances in X-men cinematic universe over the years where Magneto has used his magnetic powers on his Adamantium skeleton to shove him away, e.g. smashing him onto a tree trunk here (in X-Men: The Last Stand), or across a train coach (in the original X-Men). In these cases, while all other adamantium in Wolverine's body is bound to his skeleton, the adamantium bullet is only "lodged" somewhere in his brain, enclosed merely by tissues. Therefore, there is also a relatively-free metal piece in his brain, which should experience that magnetic force, and move inside his brain as a consequence!!

This would mean, that the region of brain cell and synapse damage would be altered, which should have some perceptible consequences, either some more amnesia, retrograde or anterograde, or at least some momentary changes/losses in bodily functions, depending on which part of the brain receives the fresh injuries due to bullet travel. And if it is memory that gets affected, it won't "heal", again as per the prior logic !!

Since Wolverine never forgot Magneto immediately after any attack, and neither is there any depiction of any bodily function loss as a consequence of these attacks, one would infer that there is no free bullet in his head, which would mean that the bullet must have gone out from the back of the head !!

• However, as Kai Qing commented below the question, this really is in the lore he points out:

  ... into the blanket of "who knows comic book physics" lore. but it should be noted that Wolverine's body, even the skeleton, seems to be able to do whatever fits the story best, including disposing of a bullet, even if it disposes it through another means aside the hole it made coming in.

I'll just add that in addition to simple ejection of the bullet itself, there may also be some other ways of assimilating the Adamantium of the bullet. After all, owing to his healing factor, his body doesn't reject Adamantium, as we have seen in the film.

So, TL;DR: The movie doesn't answer that, we can only speculate on what might have happened. In that case, (my) reasoning says it must have pierced out.

The bullet in X2 / X - Men United 'oozed' back out of his head as it did not penetrate the Adamantium coating Logan's skull where as Stryker's bullet was actually made of Adamantium, which, according to the movie logic presented in both X - Men Origins: Wolverine and Logan appears to be capable of penetrating, erm, itself, which would suggest that Stryker's bullet is still rattling around in there somewhere...