For this you need to be online and playing other people hooked up to a camera. You can check this once freeburning by looking for orange camera symbols at the 'show players' screen or by talking while online (I recommend the latter).
There are four different categories in the lineup:
1) Freeburn mugshots - Gained from takedowns while freeburning.
2) Event mugshots - Gained from takedowns in a race.
3) Rulebreaker smugshot - Gained by someone beating your freeburning road rule.
4) Photo Finishes - Gained from finishing races, win or lose (this is the only one that does not require other people to have cameras).
There is a maximum of 20 you can collect in each category.
Most people will get 40 from 1 & 4 easily enough, then struggle for the last 10. Fortunately, multiple snapshots can be received from the same driver. If you find someone with a camera who talks and is willing to help you out, you should get this fairly quickly.
Otherwise you'll be reduced to finding the game with the most camera owners in it, then setting really slow road rules on the short/most used roads in the hope one of them beats it...which is frustrating trust me. Or taking down a specific player repeatedly in every race...which won't make you very popular.
Edit: To look at how many you already have, hit start, then RB three times and you will see 'The Lineup' as the third option down.