What factors into the score you receive at the end of a multiplayer mission in ME3?
There are three things that affect your multiplayer score. Score is the same as experience in ME3, so the more you score, the faster you level.
These only show up on your screen as you earn them. Everytime you get a kill, assist, or help with an objective, a little number will pop up. (I.e., Kill +225). This one is easy to increase - just play the game, and shoot at stuff. Kill / Assist point values are based on how much damage you did to the target. You can get a "Kill" for only 80 points if you knock out the last hitpoint, but hadn't otherwise damage the target.
These show up on the end of match results screen, and are worth varying amounts of score. Every medal has differing values of Bronze, Silver and Gold, and the point values scale linearly with weapon kill medals, and geometrically with everything else. This means that getting 25 kills with a pistol and 50 kills with an assault rifle nets the exact same point bonus as getting 75 kills with an assault rifle, but getting 10 headshots and 1 grab is worth less than getting 20 headshots.
Some medals are awarded team wide. Mission completion and extraction bonuses, for instance, as well as going a certain number of waves without losing a squadmate. Like the individual achievements, these come in Bronze, Silver, and Gold.
To get your score, the game simply adds all these up.
(A more complete listing is available here.)