What would you do to prove your invention is legit?

It depends a lot on what the invention is. Usually, a demonstration is enough.

Hey. Here’s my thing. Here’s what it does.

Do it.

Don’t be lying.

Easy peasy.

Make sure it does not violate the laws of Physics. You need to be able to understand the energy balance – what energy it take to make it and how much it puts out.
Get the opinion of a reputable engineering firm. A 1 hour review would give you more info than you can imagine.

@LuckyGuy thanks for the sound advice.

I am not sure what you are trying to do obviously. But look at the system taking into account all the energy inputs and output. For example are you extracting heat from the air? or latent heat from the ground? Are you lifting something? Are you using solar, etc.
There are really good sites that explain energy and its many forms. They are helpful and will be useful when you add up the numbers.
Good luck.

@antimatter Can you describe exactly what sort of invention you are talking about? Or, is this question hypothetical (which is fine)?

Have it examined by actual scientists. Peer review. Independent reproduction of results (by scientists). Publish findings in a scientific journal. The whole thing.
Anything else would stink of quackery.

@ragingloli We use parts of Argonne national Labs’ GREET software to do energy balance analyses. Also run cradle to grave numbers using a similar tool.