2x Compression Seal,
1x Translocation Seal, can sense dungeon stairs
Itinerant meddlers, spies and general busybodies, Tinkers are well known for their compulsive mending habits and are usually welcome in any Kingdom. Until the silverware goes missing, that is.
The Tinker is a rather unconventional class. Not possessing any combat prowess, the Tinker relies on items to provide leverage. By that logic, the Tinker also requires gold to be effective. Dungeons with multiple bosses work in advantage for the Tinker, as they drop large amounts of gold for you to use. The Gloves of Midas is an excellent item to prep with the Tinker, for obvious reasons.
The ability, which allows you to sense subdungeon stairs is very useful. If a Tinker can find Good Glenrick, it equals an easy boost to level 4. A Smuggler's Den subdungeon means a stronger early game for the Tinker. Secret subdungeons are never truly inaccessable to the Tinker, thanks to his Transmutation Seals. Almost all secret subdungeons are beneficial, which is something that the Tinker should never fail to leverage.
There is no particular God that the Tinker requires to worship to be successful, but Tikki Tooki works best for the Tinker, due to the free gold you receive with Tikki's boons. If you're worshipping Tikki, consider an offensive build, so you can conserve piety.
The Tinker's potential increases as you unlock items, especially the Elite items from Bezar. Elite items cost a large amount of gold, but the Tinker comes with a free Translocation Seal, which is essentially a free item. A free Amulet of Yendor can mean an easy win.
The Tinker works with just about any race (except possibly Goblins), although you will need to have a certain strategy in mind, whether it's building a combat-oriented class for humans and orcs, glyph-based class with elves or gnomes, or abusing the Trisword or Alchemist Scroll with Halfling and Gnomes. All in all, the Tinker's a rather luck-based class, but making the most out of the RNG is what Desktop Dungeons is all about. A properly equipped (and slightly lucky) Tinker can clear even Vicious Gaan-Telet with relative ease.
Kingdom gold farming
The Tinker's ability gives him the ability to farm gold for your kingdom faster than any other class. Pick any dungeon ( Den of Danger works well), find two wall tiles, and use your free Transmutation Seals on them, which will give you 20 gold. Then, if there's a shop visible, use your Translocation Seal to steal its item. Lastly, pick up any gold or potions you see lying around. If you play a halfling, you can also convert any glyphs you see into potions. Then, leave the dungeon. Don't explore for anything more than just two walls; the idea is to get in, grab some quick gold, and get out, so you can come back and do it again in minimum time.
Once you finish your (very quick) run, make sure you press Continue on the end-of-run report. (Pressing Restart won't add the gold you gained to your kingdom's coffers.) You can start another run quickly with the "same as last run" button.
If you click fast enough, you can average at least one gold per second. Unless you're capable of routinely winning runs in about three minutes, this is perhaps the fastest way to farm gold.