Assuming the new trailer is still under warranty, I would insist that the dealer replace the tank, after first finding out the source/cause of the black gunk.
And yes, 5 gallons is way too much bleach. I usually put about a cup of bleach diluted into our 20-gallon fresh water tank. For shocking possible unhealthy mold etc., the ratio should probably be higher, but I would not go more than 1 gallon of bleach per 20 gallons, and I would not let this concentrated solution sit in the tank -- or water heater especially -- for longer than an hour plus or minus.
Be sure to run the bleach solution through all the plumbing lines including outdoor shower, etc..
And obviously rinse rinse rinse multiple times . . .
There are lots of threads here which discuss this issue.
If you use the Search menu link above right, the Google search in the middle of the pop-down box works best.