1. Each has to do his best.

2. Each has to do her best.
3. Each has to do one's best.
4. Each has to do its best.

Which expressions are grammatical?
Do I have to use only 'his'?

The best would be "Each has to do his/her best."

"its" would never be used because people aren't "it"
#3 just isn't good grammatically