It comes from the word “triumph”, which means victory. That word itself comes from Latin “triumphus” and the Roman celebration thrown for a victorious military leader.
It was shortened to “trump” while being used in card games, usually to indicate one suit having precedence over another. Then eventually it worked its way back into general language with the same meaning as “triumph”.
Edit: that’s in the context of “trump” the verb that indicates victory. For “trumped up charges”, it might come from an old English verb, “trumpen”, which means to deceive or cheat, or “trumpery”, which is pretty much a noun version of “trumpen” in meaning, though apparently it comes from a French word, “tromper”, which means to deceive (and could also be the root of “trumpen”).