I play nearly exclusively AV/AA MAX and leave the AI work to other people. If need be I can snipe people with dual Falcons. Most of the time though there are people better at the role than me, and I find maxed flak armour far more reliable against things you can't run away from.
As for the rest, a lot of it is just being in the zone at any given time. I usually suck, but occasionally get a period where my framerate is consistent and high, I've got my eye in and a comfortable firing pattern with my current weapon, and the enemies just seem to line up exactly where I expect them. These tend to lead to nice little killing sprees.
There is a serious amount of game knowledge that can help. Many bases have more or less standard lines of approach and flows of battle. Learning these just takes time, but then means you can seem to know where people are coming from before they do. Similarly there are usually ways to break that same flow that relatively few people either know or think to use.
Take this method of getting damned near anywhere you want using jump pads and LA jump jets. Essentially you manipulate the way that the game's jump pad momentum and jump jet momentums interact, letting you (with some practice) completely change the direction of flight mid air. Typically I use it to get behind (or just above) prepared enemy positions to deliver some C4 and shotgun shells.
Sadly most of my old mobility tricks were based around the old jump momentum (if you jumped while going up a slope you could get some serious air) and oddly bad framerate hurts how effectively you can climb slopes, which means I tend to rely on more basic base knowledge. Such as where there is a hidden ledge you can perch on in towers. Or which path(s) to the roof on Vanu Archives can't be easily seen by spawn camping enemies unless they want to expose themselves to your team's fire.
Only real thing I can recommend is finding a weapon you want to get good with and sticking with it for a while. Practice firing patterns (burst fire, quick bursting, etc) in VR to work out what sort of controlled fire you need at different ranges, and generally what those ranges look like on your screen. That way you can start getting your reflexes locked in and can start focusing on other aspects of play.
Alternatively just grab a shotgun on a Light Assault and focus on finding ways to get the drop on people. You want to always be engaging from the side or behind at close to extreme-close range. The shotguns are easy enough to use that you should be able to ignore the weapon mechanics side of things and just focus on movement and flow of battle. If nothing else this will teach you what approaches other people are most likely to use against you in the future.