This should be fairly easy, so no armor boost is needed.
After talking to Dwayne's ex and her new guy, don't waste time shooting her, just hop on the cycle and go straight down the road.
The key to these chase missions is to avoid crashing and just go fast enough to keep up with the him and slow down for corners. Eventually he will crash or stop. How long this takes is a luck factor of the game. With any luck, the chase/mission can be over in less than a minute.
Shooting while chasing in a car or motorcycle is difficult, leads to crashes, and brings the cops, but try it if you wish.
If he crashes, you may want to finish him with a bat or knife to avoid attracting the police, but losing them is easy enough if you prefer guns. Be sure to pick up the money before you take off.
The mission is not over until you go back to Cluckin' Bell to give the money to Dwane, who is a blue dot on your HUD.
Undress To Kill
This is not a time-sensitive mission. You should have plenty of weapons and ammo by now, but make sure your health and armor are maxed out as well.
Go in without any weapons drawn and enter the office door to the left just after you enter the club. Pull out your bat and quickly do in the boss sitting at the desk. If you use a firearm, or if a bouncer sees you going in with a weapon in hand, you'll soon have company. If you do not have a bat, it is much harder to kill him with a knife and he will probably start shooting, again attracting his thugs.
Next put your weapon away and head towards the back. After you enter the hallway in the back, pull out a firearm and do in the guard. The 2nd boss will come running out so that you can finish him off too.
Now 6-8 or so bad guys in the club will be firing at you. Take cover and kill them all. The last boss has gone to the parking lot, but he'll wait. Be careful going out the door because he may try to hit a parked car into you. Once he has done that and (hopefully) missed, he will take off. Just get in your car and give chase.
With some luck, he will not have taken off yet. Then you can pull out an RPG and end him with one shot. If you do not have an RPG yet, one can be found on a big rock off the north shore of Algonquin. Zoom in on that part of the map and you will see the rock. Drive up there, jump in the water, swim to the other (north) side of the rock, climb up to the south side of the rock and get the Rocket Launcher. It will have 3 rounds of ammo with it, but if you use all three, Niko will drop the weapon and you will have to pay $15,000 to get a new one. Also, you cannot buy more ammo for it until the Babysitting mission much later on. So don't waste those rounds.
As with all such chases, if you like shooting at fleeing cars and attracting the cops, have at it; otherwise, just keep following him until he crashes, then get out, finish him off, and pick up his cash. Head back to your crib to sleep, which both replenishes your health and lets you save.
If you attract attention before killing the first boss, all the thugs outside his office will be waiting for you with guns drawn. Kill them, then head to the back and kill one last guard there. Then you can go out the back door, but before doing so, get your RPG launcher ready and blast the car as it drives by. If you miss, he will probably get away before you can jack another car.