Identify the infinitive phrase and the subject of the infinitive in the sentence below.

To do one's best is the ultimate goal.

The infinitive phrase here is the one that includes the infinitive, that is "to do" so the whole phrase is "to do one's best"

and its subject? well, it takes the role of the subject in the sentence itself (compare "success is the ultimate goal" where the success takes this role) - but its object is "one's best".