Where did you get your version of Cave Story from?
If you got it from HERE, you would need to download the file marked: "Doukutsu Linux Port by Simon Parzer and Peter Mackay" and possibly even the 1.2 update (just drop the 1.2 version of the doukutsu_32bits or doukutsu_64bits executable into the linuxDoukutsu-1.01 directory.) .
Once you have done that, click on the file marked "doukutsu" (It has no extension), and Cave Story should run.
To use the DoConfig.exe file you need to run it in wine...
But also, in the link I posted above, there is also a native linux configure file called "DoConfigure for Linux by Sean Baker". To use that instead, you just move all the files in the DoConfigure-r2.zip to the linuxDoukutsu-1.01 directory.
and if you want to create a shortcut to Cave Story, create a new launcher and put this in the command:
sh -c "cd /your_cave_story_directory_here(linuxDoukutsu-1.01) && ./doukutsu"