This should come without much effort. Whenever you see a new melee weapon, grab it and kill someone with it; it can be an enemy or an innocent person.
Baton - Drug Bust: On the basketball courts. Some of the enemies will be holding it.
Briefcase - Run into a pedestrian who is holding the Briefcase for it to fall on the ground.
Butcher Knife - The Death By A Thousand Cuts event will give you a butcher knife.
Crowbar - In the trunks of cars. Some enemies also have this weapon.
Fish - Found in the mission Club Bam Bam.
Handbag/Bag/Purse - All 3 of these look different, in size. These are best found on women pedestrians. Run into them and they will drop the item.
Hand Grinder - Mission: Dock Yard Heist. Story related, cannot be missed.
Kitchen Knife - Some enemies will have this equipped.
Knife - Some enemies will have this equipped.
Machete - Found in mission The Election, towards the end of the level. Story related, cannot be missed.
Meat Cleaver - Drug Bust: Yen Yen Foo Underpass.
Sauce Pan - Mission Vendor Extortion. In the beginning you ill pass a kitchen with cooks in it, either pick one up from the counter or run into a cook and they will drop it.
Tire Iron - In the trunks of cars.
Umbrella - When its raining, walk around and bump into someone who is carrying one.
Eskrima Rattan Stick - Martial Arts Pack DLC.