Child's birth certificate naming both parents, father's proof of PR status, such as BRP or RC.
Complete the form you get from post office. There is no proper online application, as you still need to sign the form, partially completed and posted to you.
You need a countersignatory, who has known you (either parent) for 2 years or more, has seen your child and is a person of good standing in community. Read https://www.gov.uk/countersigning-passport-applications