Is it possible for someone to be biologically modified to be more adept at fighting? If so, what would need to be altered?

Not with current technology, but, with sufficient technological advancement? Yes, or at least it should be. Warhammer 40k’s Adeptus Astartes comes to mind as an immediate example. Also, XCOM: Enemy Within’s gene modded soldiers, so far as that goes.

At that point, the question isn’t so much “what needs to be altered?” as, “what do you want to change?”

Some of the classic examples that come to mind are increasing bone density, accelerating healing and clotting, or neural modifications. A lot of these would actually result in medical complications. Though, the complications will vary based on exactly what you’re modifying.

For example: Mast Cell production will, or at least should, accelerate clotting and wound healing. But, at the same time, that is a real medical condition, called Systemic Mastocytosis, which manifests as an increased histamine (read: allergic) response. Specifically rashes, abdominal pain, and difficulty breathing are all possibilities. So your character can’t bleed to death, but the same system means they can have a sudden fatal anaphylactic response to pet dander. (I wish I could remember where I saw someone using mast cell
overproduction to justify giving their characters near instant healing,
but this isn’t just a random example.)

So, the general plan of, “well if X is good, let’s just engineer someone to make more of it,” isn’t really a legitimate solution.

As part of a larger program, and with a lot of specific engineering, there is a real potential that you could build a better combatant. Tougher, more resilient, faster reflexes. But, as to what you’d need to change? I don’t know. It also depends on what you want out of your combatants, and just how inhuman you’re willing to make them.

Now, that’s just the medicine and science. There’s also a legal and ethical question.

The legal one is fairly simple, sort of. There’s a very real chance that biomodified supersoldiers are illegal under international law. I’ve seen this presented as flat fact in the past, but there are actually a lot of pieces to the interpretation. If you really want to dig into the legalities and bioethics of this, there’s a very detailed article from California Polytechnic on the subject.

So even if your characters can enhance their soldiers through bioengineering, it’s distinctly possible they might not be able to because of legal obligations.