For version 2.10:

 1.) Register a new name
 2.) With your new name selected, press A (unless you already chose your own set of keys, this will be the left Alt key by default).
 3.) Push your A button again.
 4.) Click "Browse".
 5.) If you are in the Zelda Classic folder, proceed to step six. If you are NOT in the Zelda Classic folder...
 5a.) Keep clicking the folder marked "..\" to go up a window until you find yourself in "My Computer".
  (Your C drive and the floppy drive will appear here, so if you see those, you're in "My Computer". One step further up is basically the desktop.)
 5b.) Find where you installed Zelda Classic, and enter the directory. (Unless you changed it at installation, it should appear right in either the C directory. You may have also installed the Zelda Classic folder in the Program Files folder.)
 6.) Select the custom quest you wish to play.
 7.) Click OK.
 8.) Click OK again, in the previous window.
 9.) Push Enter and start playing. Enjoy!