First of all, you must defeat Skeletron in your world. He is found by speaking with the Old Man in front of the Dungeon at night. You offer to remove his curse, which is summoning and killing Skeletron.
Once defeated, the Mechanic NPC can be found bound in the Dungeon. She has red hair and all you need to do is speak with her when you find her. I suggest bringing light sources with you to place as you go through the Dungeon. Once you untie her / speak with her, she becomes an NPC that can move into a suitable house.
The Mechanic sells the wire and wrenches you need. You can purchase any color wrench you want. Each one just lays down a different color of wire when placed. You need to place 100 wire with any wrench. You can do this by placing the Wrench in your top row and using it like a weapon with RT. You can place wire just about anywhere, you do not need a wall or surface. As long as you have wire in your inventory (it can also go in your ammo slots), you can place it with wrench. Wire is used to link together various items, like switches and actuators to open doors, activate traps, turn on lights, etc.
Wires can be harvested from the world using a Wire Cutter, but it easiest to purchase them from the Mechanic. I am fairly certain you cannot find a Wrench in any chest, you must purchase one or get one from a friend's world.
You only need to place 100 wire with ANY wrench for the achievement. It does not need to be connected to anything, or be a continuous line.
Grab a wrench (only sold by Mechanic) and 110 pieces of wire. Then run back and forth holding down RT with the wrench selected until achievement pops.