I think if you'd like to keep the phrase structure, I'd say you could go with "I would put in my best effort" or "I would put forth my best effort". you could also use "make an effort" like what Jmat suggested, but if you say "I would make an effort to do/be ___" it does not quite feel the same as "I would do my best" because you could make an effort to do most things, but making an effort is not quite the same as giving it your all/your best, in my humble opinion

Usually when you use "make," you would say "make an effort" and not "make my effort." For your case, this sounds better: I will give my utmost effort to be a...

Even though "to do an effort" makes sense logically, people never word it like this and it sounds awkward. If you do say this, people will still understand you. Using 'make' here might seem counter-intuitive, but "to make an effort" is a phrasal verb. There's really no reason that 'make' is used here, except that it's convention.