Its just gonna take practice. The best suggestion so far is to pick up a copy of unreal tournament. Its fairly representative of FPS games, as far as deathmatch goes.
You're going to have to learn a lot.
First you need to get comfy with the KB and mouse together. It has to become total second nature, where the mouse and KB just does what you want by thinking about it. That'll come with time. I shunned the mouse/KB for years cause it was so hard at first, but then one day I just decided to start doing it and stick to it, and now I shock people with my mouse skills.
Then you're going to have to learn some proper tactics. Circle-strafing, sidestepping, shooting the ground for splash damage, evasive manuvers, camping (yes, its a skill), ducking, some games require slowing down/standing still for proper aim, hiding while you reload.
And eventually you'll just get your own personal style. Some go out blazing and get kills from the element of suprise, some choose to stock up on weapons and armor by winning through superior firepower, some people are scavengers, picking on the weak, and some are just bastard campers that sit and wait for you to walk by while they pick you off with a sniper rifle from a mile away. The best players know these styles well, and choose appropriately consdering the situation.
Natural talent helps too.
You just need to try and pick the best possible course of action considering your current strengths/weaknesses. (Health, Ammo, Weaponry) Try and keep an eye out on the enivronment, and learn your enemy. This applies to both single player and multiplayer. Everyone/thing has a weakness, and you need to exploit the hell out of it.
For example, if he's throwing rockets at your face, try your best to get in his. His rockets are useless at close range, and are as likely to get him killed as to get you killed. If you're up against a sniper, dont stand still, and dont run in a straight line. If youre zig zagging towards him, he has almost no chance of hitting you, and I'm amazed at how few people understand this simple concept. Thats the single most important thing. DO NOT STOP MOVING. If your accuracy relies on standing still, dont do it for too long.