How to renew your passport in Mexico ?

Passport for what country?

Sounds like this boyfriend is an online one you have never met. It also sounds like he is trying to scam you.

A birth certificate is not required to renew a passport. And unless this guy is Mexican, then their government does not get involved.

Please do not send this person money, regardless of their sob story and how much they claim to love you 

Do you know your "boyfriend" from real life?

Your bf needs to apply for renewal of his passport through his country's nearest consulate to where he is staying.

If he wants you to give him money, DO NOT! Let him deal with his own issues.

If he's in the US and trying to renew a Mexican passport the fee is $74 for three year document and $101 for a six year document. Perhaps he's trying to shave a profit off of the money he says he's borrowing from you but likely will never pay back. 

The cost of a Mexican passport renewal in Mexico is $65.  Renewals do not require a birth certificate.

According to the Mexican Embassy's website, a three year passport costs $74.