There are only three bounty hunter missions, and the MO is the same for each- find the target, shoot them with the dart gun Marshall gives you, then dump their body in the trunk of a car and drive back to Marshall. At least two of the bounties will try to run- they're about as fast as Lincoln, but they'll occasionally stop to catch their breath. Which is good, because the dart gun is wildly inaccurate, and you'll need to get close for a sure shot.
The other thing that's easy to overlook is the time limit. From the moment Marshall gives you the mission, you only have a few minutes to drive over and get eyes on the target. The time's not so tight you need to break out your racing skills- just don't get distracted with other things on the way.
Bounty #1 Ray Duplass
Something of a training mission, this. When Marshall gives you the op, you have seven minutes to reach the objective marker. When you arrive, exit the car and Lincoln will lean back and wait for his target to show. When his target does just that, give chase on foot, taking shots whenever Duplass pauses, or is just close enough to make a good target. The crowd here is too distraught for anyone to worry about calling the police, so you don't need to keep holstering your gun while in pursuit.
Once you've got him, drop in the trunk of the nearest car (little trunk-insert icons will appear on eligible vehicles).
Then drive back to the compound and inside garage 8 to complete the mission
Bounty #2 Kurt Mancini
As before, you'll have seven minutes to reach your target once Marshall gives the mission. Don't dawdle, though- your first lead won't pan out, and you don't get any extra time added to the clock while driving to the second.
First stop is Mancini's hotel room, but he isn't there. Examine the note on the small table across from the bed for a clue to his current location.
Drive over to the French Ward objective to trigger a cut scene where Mancini make a break from his captors. Do *not* engage these guys- the cops will be all over you in seconds, and chasing down Mancini is irritating enough as it is. Instead, just keep jogging after your target- you'll soon outdistance the goons and, once Mancini runs into a relatively secluded area, you can take him down as per usual. Stuff him in a trunk, but try to pick a car with a bit of speed, because on the drive back you'll be hit by a retaliation squad.
Granted these guys are in pretty clunky trucks too, so there's no reason to engage if you don't want to. Just keep one step ahead and pull into the garage as usual to end the mission.
Bounty #3 Brian Patrick O'Malley
Your final target. This time there's only four minutes on the clock, but there's no extra shenanigans to account for either. Drive to the bar where O'Malley is holding court. Slip around to the back and jimmy the door to find someone trussed up in the kitchen.
Free him, and get your dart gun ready, because O'Malley will immediately enter with a shotgun. Dart him, then kill the other hostiles.
When you're clear, just trunk & dump (there's a nice Lassiter Courant in the parking lot). Drop him off at the compound then talk to Marshall one last time to complete the Bounty Hunting quest, and unlock the Stiletto as a deliverable car.