On someone's trail phrase
The expression can be rewritten as "on the trail of someone".
If a person is "on his/her trail", he/she is following, tracking someone in order to find them or find out about them.
He must have done something wrong or even illegally. That's why the police in on his trail.
A: "What are you doing here?" B: "I'm on that girl's trail. I have found something abnormal from her computer. She's a suspect in a case".
A person who is completely unknown or has no connection with you
The guy I talked to at the prom last night is a perfect stranger.