Hello there, I have a question on the circuit breaker for pump. The pump that I'm using is 2HP with power supply 240v and the ampere would be 6.25 am I right? What is the min circuit breaker ampere that I can use?

6.25 amps sounds about right.
You can use a 15 amp circuit for this if you want.

Does the pump have initial high amp during start up? How to clarify the amp for the pump when it is running?

All motors have a higher startup amperage, but circuit breakers are designed to handle this.
The only way to clarify is to look at the plate on the motor.