Are there any tricks to Rival Smash?

In general, you should use a character that has several 1-hit KOs and can survive a decent number of hits from your rival. Heavy characters, therefore, are better generally. Then follow these general ideas:

• Spam your 1-hit KO attacks
• Mostly ignore your rival, but when you can hit your rival safely, do. Mostly, I find this easy to do with an up-smash attack when your rival is on a platform directly above you.
• Use the items when they appear if you can get to them safely (the pow block is especially good for driving up your score since it is multi-use and can knock out multiple opponents at a time).
• Stay alive!

Specifically, I found it very easy to get high scores with Peach. This is mostly because the computer is particularly bad at this challenge with Peach.

• Only a few of your attacks are 1-hit KOs. You will be spamming these while the CPU performs slow, non-1-hit KO attacks.
• Peach has very high survivability even for newer players due to her ability to float long horizontal distances.
• It takes the CPU forever to return to the stage when knocked horizontally. The CPU typically exhausts the entire floating air-time of Peach before grabbing the edge, allowing you to rack up the score.