First, and most obviously, make sure joystick is calibrated. Do this in Windows>Control Panel, as well as in BFHQ.
Second, are you cutting your throttle? Ease back on the stick to bring the nose level, and cut your throttle to half, or as low as 33%.
I disagree with many above posters, I think hovering is essential. You need to hover for repairs on a pad or box (if you land, you are even MORE of a sitting duck (can't move) and also SpecOps can now kill you). And you need to hover to set up your gunners.
Single player practice: Attack chopper spawns on Zatar at Centre Island. Cap it, then fly back to your main for practice. Operation Clean Sweep, BH spawns at USA main. At first, "teammates" will fly away with it, but if you do not get in, and just stand around at pad, they will soon die and BH will respawn. (Note: The enemy AA stingers & vehicles are WAY better in single player than online play, so don't engage the enemy at first or you will get killed.)