The bike parts are extremely easy to get when you know where they are. You will be able to get all parts straight after the pawnshop except for the handle bars, because a surivor has them (Jason Wong). After 3:00 PM you will be able to find him and the handle bars. Read each description for the locations and conditions for each part. They are listed in order as you should grab them. When you get on, you will have to return it to the safehouse and put it on the bike. You can press to attack with it, since it's two handed and you can't switch weapons. It cannot break. To see a list of bike parts, hit and select Bike Parts.
Note: You can finish all the bike parts and finish up everything else after building it. It will not end your playthrough if you fix it completely.
1. Wheel: After unlocking access to the pawnshop, by saving the owner, you may buy the wheel for $5,000. You won't have enough at first, so run across the street to the casino and collect the money there (bashing the slot machines for extra). You can now buy it.
2. Gasoline Canister: On your way to the safehouse from grabbing the wheel, stop by the gas station by the pumps. There's a red canister there. Pick it (dropping your Wheel) and then pick up the wheel again. The canister is a one handed item, so you can take both to the safehouse at once.
3. Engine: The engine is in the alley next to the movie theatre. Go through the movie theatre and out the exit into the alley (this unlocks a door for you). It's on the ground.
4. Bike Forks: This item is locked in a shed, located on the South East part of your map. It's the small building next to the grey circle (which respresents a giant tire sign). You must first get a key, which is located on the second floor of the hotel. To get there, go into the alley inbetween Bob's Fish 'n Hunt and Still Creek Casino. There's a green garbage can at the end. You can use this to jump to the fire escape (have to time it right). Now go up into the hotel. Go left and then right at the broken stairs. Inside the lone room there is a key to unlock the shed (gives you an unlocked door).
5. Handle Bars: These can only be obtained from the survivor Jason Wong. He will appear anytime after 3:00 PM behind Uncle Bill's Department Store (exit out the back). He will want a broadsword before he gives you the handle bars. You will want to get the sword before you meet him. To obtain it, jump up to the building Bob is on (yelling/shooting guy). Now face the direction of the other building and jump to it. Go to the other side of it and go left along the awning. Now you can jump up to the hunting store, and drop down through the hatch. The sword is inside, and you'll get an unlocked door out of it too.