Need toilet advice

I got to go back to my place (my mom's house) and we are incredibly lucky re: damage

However, when we went to fix what we could, I noticed the toilet upstairs had really dark/gross water. I'm assuming some sort of a backup occurred. Is there a way that I can deal with this safely?

We're bringing water (not bottled) for hand washing and toilets, so can I just fill up the tank and flush it down? Is there a concern about this? The last thing that I want is overflowing sewage upstairs.