1. Get good FPS. As much as it sucks to hear, the hardware you have DOES play a major role. IMO, under 60fps is basically useless in a competitive FPS. I prefer not to use Vsync, but instead cap the FPS via the in-game console (vsync SEEMS to add a bit of mouse lag), but you can test that for yourself. Personally, I have FPS capped at 120, and my settings set so that I ALWAYS get 120fps (in BF3). Remember though its not JUST FPS. If you're getting 120fps, but it stutters a lot, its still just as bad as playing 20-30fps. The goal is to get an extremely smooth 60fps+. Try to set the fps CAP so that the FPS doesn't fluctuate massively, as that can make it seem like its getting 'sluggish', even though the FPS is over 60.
2. Good sound. Again, hardware does matter. A good headset is pretty helpful, but you dont need to go all-out. A $50 sennheiser headset is 90% as good as a $200 FOR GAMES (audiophiles, I know there is a massive technical difference). However, if you have the cash, the Sennheiser PC 350's are an AMAZING headset. Remember though, you'll want something to drive them. A good sound card can help as well, as it simulates direction much better than the default mobo crap. Again, you can get them between $50 and $200. There IS a difference, but its not insanely huge (again, I recommend the top-tier Creative X-FI Titanium HD).
3. Good input. Last hardware bit, I promise. Input is also key. Dont play with a default DELL mouse. Pick up something good. Be it a mid-range mouse, or some Logitech G500 thing, it WILL make a difference. Sadly they've stopped making the Logitech G5, so your best bet (IMO) is the Logitech G500. Don't go wireless either. Some people say the cord type matters as well, but I doubt it. As long as the cord isnt as stiff as a stick, it'll be fine. Also a keyboard CAN help if what you're using is crap. Again, a $30 upgrade can make a huge difference. I have no idea what the technical term is, but however the keys work in the FIRST GEN Logitech G15 is ideal IMO.
4. Do some aiming practice. I remember about a year after CoD4 came out, I decided (for no reason) that I wanted to be good with a bolt rifle. I set myself some random targets at medium range (15-20m, targets like "top of the door" or "that pole") and would try to shoot them after turning. So for example, i'd strafe past an open window, and try to hit the target without stopping. After a good 15 minutes my skill had gone up a fair bit, and I could actually play fairly well with just a bolt-rifle. A few weeks later, and I was usually top of the leader boards.
You're probably not going to do stuff like that in BF3 (doesnt REALLY work), but that kind of stuff still helps. Go into SP or empty MP servers, pick random targets and hit them as fast as possible. Try landing a 360 in a jump (run forward, while going forward jump and turn, land facing the same direction), that kind of stuff. Anything that requires mouse accuracy. After that, its really just play a whole lot.
Try and pick one weapon though, as each weapon acts slightly different. I would recommend the UMP45, MP7, PP2000 or any of the Engie weapons. Play with that till your KD gets around 2.0 per round, and then try some other stuff.