The main source of income is winning runs. You get kingdom gold for:
Money and items your character carries out (you get these even if you chicken out, as long as you don't get killed, but you don't get much from this)
Completing a quest (usually several hundred gold)
The boss trophy (diminishing returns - price goes down the more of the same trophy you get, but starts around 300g)
Completing a dungeon with a new class (50g each time)
Obtaining a badge for the first time on a dungeon (10-50g)
As for how to win runs while you are still learning the ropes, one beginner strategy is to run Hobbler's Hold repeatedly, as it's a lot easier than a normal game. Once you can win that dungeon with ease, move onto the normal dungeons. (This effectively locks out Hobbler's Hold as a source of income, but don't worry, you'll get much better returns from doing quests and so on in the normal dungeons.) Once you are able to complete 'normal' difficulty runs, you should have plenty of money coming in.
It's normal to lose a lot of runs when you're starting out. Although it's a 'quick' game in that each run only takes a few minutes, Desktop Dungeons is actually a difficult and complex game that takes a long time to master.