I have never been an efficient killer, but a few tips I got from others helped me score better:
1) Realize that any moment not damaging the enemy is an opportunity wasted. Position yourself overlooking a spawn point at the beginning of each wave and get shooting/casting as quickly as possible. Listen carefully for enemies so you don't wind up running around not finding anything to kill. Seek them out instead of waiting for them to come to you.
2) Aim for the head. I'm not great at this, but if you can kill even 20% faster by getting the occasional headshot it's worth it.
3) Kill mooks instead of wasting time wearing down that atlas that someone else is going to snipe at the last second.
There are less savory ways to increase your score as well, such as overwriting others' primed enemies with your own primer so you get the credit when they detonate, etc., but I don't think they're necessary. Also, douchebaggery isn't fun.
These are all good tips, especially #3 - you will waste a lot of time going after bosses when you can be racking up points by mowing down mooks. I'd add:
1. Use classes with AOE based attacks. If you look at the damage a power does, you can multiply that damage for every additional enemy you hit. Obviously, if you're willing to carry Grenade Capacity Gear and burn a bunch of Thermal Clip Packs, you can score a ton of points by burning up spawns with grenade spam. But if you don't want to do this, biotics can score pretty high without a ton of skill too. Besides the fact that you get more points (I think) for killing enemies with a biotic combo, you are also getting a lot of points for damage you're doing to other enemies in the explosion's range. Plus, you will rack up lame assist points too.
2. Use Geth Scanner. Other than Grenade Gear, there isn't really any other gear that is going to make you better than this one. You will get to enemies (and thus kill them) quicker and you will help yourself die less by knowing where all the enemies are.
3. Use the Geth Infiltrator. Seriously, it's really hard to score low with the GI if you're using a powerful weapon you are comfortable with handling. I always default to the Piranha with Extended Barrel, Choke, and APIV because it's so easy to use, but any good gun will do just fine.
4. Save missiles for the extraction wave. You should never use a missile before Wave 11 unless it's necessary to avoid wiping. Then, run from spawn to spawn on Wave 11, taking them out with missiles. You will rack up serious points quickly, and it has the added bonus of causing sync-kill units to spawn behind your teammates, hopefully taking them out and preventing them from getting extraction points. Win-win! 