Difficulty: Any
There are 5 different melee weapons that you'll need to acquire, upgrade, and use until they break.
Weapon: Pipe
Location: Bill's Town - The Woods
First upgraded weapon in the game and can be found on the root of the small building you need to lay the plank on to cross.
Weapon: 2x4
Location: Bill's Town - Graveyard
You can sneak pass some Clickers around the crypts in the graveyard. You'll eventually come to an area with a found and a few stone benches around. The 2x4 will be on one of the stone benches.
Weapon: Baseball Bat
Location: Bill's Town - Graveyard
Once you have dropped down into the yard with the pool from the treehouse, go right into the house then head left to the garage. You'll see a shelving unit with a baseball bat.
Weapon: Hatchet
Location: Tommy's Dam - Hydroelectric Dam
You'll continue through the dam with Tommy until you are invaded. You'll make your way back the way you came and come outside. The hatchet will be laying near a crate. Almost the exact location where you can pet Buckley on the way inside.
Weapon: Machete
Location: Tommy's Dam - Ranch House
You and Tommy will go on horseback to find Ellie and eventually come to a stop at a ranch house. You'll find Ellie upstairs. The house will be invaded and there will be two enemies on the top floor. One will be armed with a gun, and most likely the first one you'll see, and then the other will be hiding with a machete ready to charge you. Once you drop him, upgrade it and kill the other 3 guys downstairs.
There are gaps in between the weapons, so if you find yourself without the materials needed just hold onto them until you do find some. The first weapon should already be upgraded.
Hope this helps!