This could obviously be done with flash/actionscript and there is likely a few players out there for this. It would be relatively simple to code such a thing though with basic controls and you would merely need to check if the current song has reached the end, possibly have a short delay and then play the next song in a list.
If flash isn't your thing HTML5 is fully capable of doing this as well. Though you would cut off your system to anyone not using a modern browser but that's your personal decision.
When I get home later tonight I could always write up some code if you somehow feel you can't research any of this or just have no clue on how to begin coding either solution.