It is a coin a liberty head dime with no mint mark or date on it and has 999 fine silver one troy ounce on the back and a balance a fake?

My guess is that it's not a fake. If it were a dime it would say One Dime, but it instead shows silver fineness and a weight, so what you have is almost certainly a bullion piece that bears a copy of the dime's design.

These items are sold as a way for people to easily own small amounts of precious metals. Right now silver is selling for around $13 to $13.50 an ounce so your bullion piece would be worth somewhere in that neighborhood.