Know the levels. You'll need to become familiar with the layout of the maps if you expect to be successful on them. Learn where the weapons and powerups are, and how to get from one place to another quickly. Until you do, your enemies will always be one step ahead of you. Use sounds to your advantage.
Sounds can easily give away locations of other players.
Explosions, footsteps, jumping, items being picked up, and splashing can all give you clues to where the other players are. Look for clues in the environment. If items are missing, someone has been in that area recently . If you see a platform or elevator moving, someone's nearby. Take a step back when opening doors.