First, you'll need to open Big Picture mode. The button to do so is in the top right corner of your Steam client.

Next, go to Big Picture mode's options, open controller settings, and enable configuration support for your controller.

Now, go to your game library and click on Drunken Wrestlers 2. Click on Manage Game, then Controller Options, and turn on Steam Input per-game setting.

Finally, go to Controller Configuration. Press Square / X(if neither work, look at the bottom of the screen and the button will be listed) to open Browse Configs. Then, go to the Community Tab and select Kickboxing. Press Square/X again to apply the config.

You can now open the game and play with a controller!

Be sure to review the configuration to learn the controls, and edit them how you like.

By default:

Left Stick / D-Pad - WSAD
Square / X - Left Arrow
Triangle / Y - Right Arrow
Cross / A - Up Arrow
Circle / B - Down Arrow
Right Trigger - Space
Left Trigger - Left Shift
Left Bumper / Right Bumper - Left Alt
Options / Start - Esc
Share / Select - Tab
Right Stick / Right Stick Click - Mouse / Left Mouse Click
Touch Pad - Scroll Wheel