You shoud first create a good (yet simple) model of your TileMap. I am thinking of a Map class, and it could contains an 2D-Array of Tile objects, which have an Image property.

This is the basic model, you can later add properties for supporting overlays, checking for walkable areas, jagged (no square) maps and so on.

Derive from JPanel or JComponent, and override its paint-Method. Check for the JavaDoc of Graphics class. In the method you go through the tiles of the map and draw its images (using Graphic's methods).

With a MouseListener attached, you can check which Tile you have clicked on, change its Image and draw the component anew.

Try to play around a bit with it, the details depend on the style of you map and game.

Why don't you want to include a plot? Setting seems interestng and fresh...

Anyway i would add something unique to a mythical name, just to avoid competing with wikipedia on google search.

Something like "Canaan Carnage", or "Uwe (the hero) whips the Ugarits"

I dont think Java can load the Windows-specific .bmp files. Try to convert it into a gif or jpeg if loading it via ImageIcon.

Tried the game, found no bugs (made it to level three or four), and enjoyed it much
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I especially like the balanced design for a rather small game like this: just start playing with no long set of complicated rules but an intuitive controlling. Keeps the frustration level low at the beginning. If you go on an evolve it, i would let other items that apples and weapons introduce step by step as you go further down the dungeon.

Also very clean programming i think.