Unlocking all Seven Deadly Weapons is not all too difficult. Once you have each one simply get 25 kills with each. I used them one at a time so when I got the status pop up I would know exactly when all 25 kills were done for each weapon. The weapons are as follows:
Armchair-A-Geddon - This is the easiest to unlock. There will be a mission called "Take A Break" where you need to go find a place to sit down. Select the mission and follow the waypoint to the top of a building where you will find the chair.
Gallows Dodger - For this you need to complete the Dex side event. The Dex will randomly show up throughout the world and will be marked with a red target. Once you kill 7 of them you will earn this gun.
Last Supper - This gun requires that you simply visit each of the three drivethru locations in the game. If you reach level 15 you will have the option to unlock the collectible finder which will make this a breeze. The 3 locations will have a toilet icon.
Uriels Edge - Complete all 5 of the Marshalling Grounds spread throughout the world. Once you complete the 5th one you will unlock the ability to travel to a secret room where the sword is located. It will also pop up as a quest so you shouldn't miss it.
Boom Chicka - For this you need all 5 of Blackbeard's chests. Each chest requires that you find 3 hidden glyphs to unlock it. Again if you use the collectible finder you can easily see all 15 glyphs and all 5 chests on the map.
Diamond Sting - Buy from any weapon shop for $100,000. It is in the SMG category.
Ark of the Covenant - This can be earned by killing 10 Archdukes and unlocking the top agent, Professor Genki. If you're roaming around town and get your notoriety level maxed out an Archduke will appear. After ten Archduke kills Genki will appear with the weapon which you will get upon killing him.