I'm having trouble finding the goblin tinkerer
The Goblin Tinkerer is not in a specific location. He will simply spawn near you at random, then respawn later if you miss him.
The Goblin Tinkerer can be found anywhere on the Cavern layer of your world after you've defeated a Goblin Army invasion. However, like all bound NPCs, he is not placed in a single specific spot in the world for you to find; rather, as you're moving around the relevant area, the game will randomly decide to spawn him somewhere near you, and you may stumble upon him.
You are not required to search inside the Dungeon. However, you may have better luck looking there (at the Cavern level) simply due to the abundance of easily-accessible open space in which the Goblin Tinkerer might spawn; the Dungeon does not inherently increase the chances of finding him. Items which might help you in locating him (and other bound NPCs) include the Hunter Potion (which will highlight bound NPCs in addition to enemies) and the Lifeform Analyzer (which will alert you if certain rare creatures, including bound NPCs, are nearby). Unfortunately, as of the writing of this answer, the Lifeform Analyzer is not yet available on Mobile, but the Hunter Potion is, and likely just as good (if consumable).