I would like to seek legal advice on how can I change my son's last name to mine. I got pregnant 6 yrs ago but my then partner are not married and have since long broken up. Nonetheless, he signed the affidavit of acknowledgement of paternity, thus, my son has used the surname of his father ever since.

I would like my son to use my surname instead. What legal steps should I make & how long is the procedure?

It depends on the state you are in. If NY, you go to the courthouse and apply (or start the process online), but without the father's signature of approval, you have to go before a judge to explain and hope it's granted.

https://www.nycourts.gov/courts/nyc/civil/int_minornamechange.shtml

The father will need to be contacted and his agreement may be needed. It depends on the local courts. I would advise consulting a local Family Law attorney who will know the courts and the process.