The 'already' is a bit redundant with the 'before'. I think just "I've come here before." or "I've already been here." would be more natural English.

Good English doesn't always have to be concise. Only technical reports.

As a native speaker, I don't see any problem with this phrase. The repetition, though present, is not egregious.

Just because you're a 'native speaker' doesn't mean you can't be wrong. I suggest you get a second opinion, or trust Google.

"I've already come here before." - One Google hit (that isn't yours) and it's translated from Vietnamese.

"I've come here before." - 81 hits

"I've already been here." - 454 hits (but about half are false positives).

Again, I don't see any serious issue with this sentence. There's no great flaw and it's not grammatically wrong. The repetition is used to emphasize the fact. If you like, add an alternative translation and add a "repetitive version" tag or something of the like.

"I've already been here before" yields 28 results on Google, and so the "already... before" structure may not be as wrong as you think. Let's not eliminate the minority here.