Learn the points where people can shoot you (typically, chest, back, shoulders, sides of your gun and straight down the barrel).
Keep moving – shots can take a second or two to register, so if your opponent isn’t good with his or her aim, you can keep them from scoring a kill. Unless you care about your shot/hit ratio, or unless the game actually has “limited ammo” (unlikely), you have unlimited ammo, and it’s perfectly acceptable to keep shooting until you see your opponent’s suit go dark / you score a kill.
It may help to keep your back to a wall when possible (including while you move). That’ll reduce the number of potential points people can shoot you with.
Typically, once hit, your suit and gun is disabled for ten seconds (a cooldown time). This give you time to move to a better location. Or alternatively, stalk someone during this time, as your suit will probably be dark and harder to see. On the flip side, you could be mean and just follow someone you just hit, so you can get them as soon as their suit becomes active again. You could probably earn more points finding others during that time though.
If the layout has multiple levels (you can go up onto structures), go up. People have a tendency not to look up. So if you can find a good location where you can see a decent area, stay there for a bit and just snipe anyone that passes below you. However, make sure you’re aware of the entrances and exits to the upper levels as well as windows behind you and grates below you where you could get shot. This is also a good place to practice your aim – if you can aim and shoot faster than your opponent, in a shootout, you can probably take him or her out before they get you.
Also, scout out the layout as soon as you get in. If possible, try to be one of the first people to get into the arena – that’ll give you time to just walk around and find a good place to shoot people from.
Wear comfortable clothing and shoes you can run in. You’re not supposed to run, but people do it anyways (or powerwalk, anyways). And the more sneaky you can be, the better.
Most importantly, have fun. Don’t complain when you get shot and try not to gloat too much when you show up at the top of the list. 😉
Note: These hints are based off the assumption you’re playing Team Deathmatch or Free for all styles (where the goal is to rack up as many kills as possible, either for your team or for yourself, while avoiding being killed). It may not be as applicable for other game modes, such as Capture the Flag.