Screw kill count. Kills only look at your last hit on a champion. What matters is how much damage did you do overall, and did you stun them at the right time.
My Ryze has absolutely horrific kill count (maximum of 6 in a game where we ended with 40-10). My Karthus killed 7 people in my last game (final score 50-10), Rammus killed 22.
Overall damage--I lead Rammus by about 3 times. But he has 3.5 DoT ticks (2x Sunfire + Tremor, and he can taunt people onto his DBC), so he has a much better chance of landing the killing blow than me. But like I said before--I triple his total damage count, and quadrouple his assist (I was there for every one of his kills, but he usually isn't there when I tank for the rest of my team...as Karthus).
So screw kill count. If you do damage, and the opponents die to your team anyway, then it's good enough. Sure, it's important to be able to finish people off as DPS, but if there are other people who simply were "stealing" your kills (intentionally or not), then don't worry about it. But if you're the only DPS on your team..then it's a real problem.