Ahhh, good old Resident Evil. I have played through the original PSone copy I think maybe 5 times; RE2 at least 6 times; and RE3 maybe 6 times as well.
As others have said, Jill does find more ammo and also has the lockpick (she is the master of unlocking after all). Inventory space is the same between the two though. And the majority of the game itself is the same as well, as far as rooms and puzzles goes. RE2 is where things branched off more between Leon and Claire.
That being said, you shouldn't have any issues playing as Chris. Zombies do not respawn if you kill them "for good" which is when the pool of blood shows. There should also be a notable, last grasp groan then that happens. You can definitely kill the vast majority of enemies as well, but not all of them.
1) Aiming. Sounds dumb but because of the tank controls, is easier said than done. You will get better with practice though. Taking more time in between shots, and letting them get closer to you, will up your chances of getting a headshot, which does massively more damage. This usually isn't a problem with only 1-3 zombies; just be confident in yourself letting them close in.
2) If you are able, especially with only one zombie, after you knock it down the first time, run up to it and hit a few times with your knife. You are able to aim down at their bodies. If it starts to get up, then move back away again. A lot of times though, depending on where your bullets landed before knocking the zombie down, you can finish them off this way, saving ammo.
3) As mentioned above, they are dead for good once you see the pool of blood.
4) Depending on the sequence/area, it is not worth shooting at all. The dogs and crows come to mind, as you should be able to grab the very few (if any items) in areas while running past them. Other areas though you will realize you have to go through many times (RE1 has a ton of backtracking, easily the most in the series) so in those hallways you might want to get rid of the enemies just to make things easier.
5) Saves are limited, so be mindful of that. I never really felt too constrained by this though. Other than one boss fight, where I had put myself in a corner with the save file, ammo count and health items, and essentially had no choice but to beat this particular boss almost perfectly with no wasted ammo or getting hit. Took me at least a dozen tries lol
Good luck and enjoy, it is one heck of a great game!