How do I find the goblin tinkerer

You should wait for a while, I would recommend you to pretty much stop searching for him, it's easier to find things when you aren't looking for them.
 
I recommend using a Lifeform Analyzer. They can be bought from the Traveling Merchant for five gold, though he's only selling them sometimes.
 
You make a lot of threads. I'd refer to the Terraria Wiki, or just find out through acquired knowledge from more play-time.
 
The Goblin Tinkerer kind of "follows" you around. He won't be in any specific place in the world so Tedit won't help.

After you have completed the Goblin Army event, the Bound Goblin will spawn somewhere near(ish) you when you are underground, and will despawn if you go too far away from him, respawning again near you after a couple if minutes.

Don't bother trying to search for him over a wide area - that can actually be harder as he constantly despawns and respawns as you travel the world. If you can, find a fairly open area somewhere underground, that doesn't have too many closed off caves. If there are, open them up so there are no places enemies can get trapped, that way he won't appear hidden in a cave you can't see.

Using a water candle will increase the spawn rate of NPCs as well as enemies, and as has been suggested, hunter potion will highlight him. If you can get a life lifeform analyser that will help immensely - it will say a "Bound Goblin" has been detected nearby.

Just stay roughly in that area killing any enemies and he will show up. If he doesn't, just leave the area and come back again, and he will eventually.
 
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