You gain experience for just about everything you do in multiplayer for Battlefield 3. Here are a few of my favorite tips to get points even if you aren't always on top of the leaderboards for kills:
Play Support class and throw down ammo. Try to join a squad that has snipers, then spawn on them and immediately toss an ammo box. They almost always give points.
Stick to one gun in support and try to get enough kills with it to unlock extended mag. Equip the extended mag along with the suppression perk, then just open fire on any enemies you find. You may not get the kill, and you may end up being killed instead, but chances are great you will get both a kill assist and a suppression assist once someone else kills your target. Those two numbers combined almost equal 100 points anyway. I like to use the guns with higher firing rates for this method.
Play Assault class and throw down medkits. Don't be afraid to revive with the defibs as well, tho you may make people angry if they didnt want to be rescued.
Spot any enemies you see, its worth points if someone else kills them for you.
Play conquest and just capture enemy bases for easy points.
Jump in a tank or helicopter that has a skilled pilot. You can spot enemies while sitting in there, and you will rake in passenger kill assists.
Throw four C-4 packs on an empty vehicle. If you notice the vehicle disappear on your mini map later, chances are an enemy just jumped inside it. Switch to your detonator and click the trigger, it should disable or destroy the vehicle and cause some kills as well.
**NOTE** some servers will specifically mention the C-4 method below is not allowed, be sure to follow the server rules to avoid being kicked:
Put C-4 on the flags then watch your mini map on the lower left side of the screen. If the letter of the flag you put C-4 on starts to blink, switch and detonate your C-4. It may kill the base attackers remotely.
Play team deathmatch on the Canals map, and hide in the crates area. Throw medkits or ammo packs in the middle of the crates, teammates will heal or reload as they run past which gives you points.
Another great trick someone else showed me was to play Recon on Canals, but take a SMG or shotgun instead of a sniper rifle. Get in the middle of the crates and find a cozy spot to put down a motion detector. You will rake in motion assist kills, and if anyone comes around your corner just open fire on them.
Keep your eyes open for double XP events and play as much as possible during those times.
If you notice you are on the winning team more often than not, try sending friend invites to the other players on that team and playing with them any chance you get.