Here is the full list of all 10 Bike Rush achievements.
Destroy 12 cop vehicles by using handgun
Destroy 5 cop cars using grenade (in sequence)
Destroy 12 cop vehicles by using grenades
Destroy 2 cop vehicles using mine (bomb) bonus
Destroy 18 cop vehicles by the end of the race
Complete the race with at least 3 respawns left
Destroy 20 civilian vehicles by the end of the race
Destroy the boss armored SUV using grenade
Don't let helicopter to hang on you more than 2 seconds
Get 35 000 points by the end of the race