A cliffhanger is where you knock your opponent to a lower level of the stage, but they cling on by their fingertips (leading into a cutscene animation). All you need to do is let your opponent do this to you, and block as they attack you shortly after.
Start a single player VS match and choose your characters (it doesn't matter who). Make sure you choose "Scramble" (the first version) as the stage.
You see the pile of girders hanging from a crane at one side of the stage? That is where you want to be. Position yourself with your back to these girders as soon as possible and let your opponent knock you into them. This will make the crane dump the girders loosely onto the edge of the stage. We are now set up perfectly. Let them do it a second time and the cliffhanger will be triggered.
Spam the Block button as you are knocked towards the edge (Thanks XenoBlade) and you should cling to the edge by your fingertips. Then your opponent will run at you to attack you. While clinging to the edge, simply hold the Block button and you will easily block their attack. The achievement will pop shortly after you land.
Here is a video I made which might help.