When the birth of my daughter was registered at the amphur, i was present and had to sign copies of my passport. Does this mean they accept me as the legitimate father? The mother-in-law is named as the person notifying the birth and i'm not married to the mother.

Congratulations

Unless you have registered the marriage to the mother at the amphur, No. The mother has sole custody and you have no rights or obligations to the child

You will either have to register the marriage or legitimize at amphur (when the child turns approx 7, all 3! parties must agree) or do it in court. Court is standard procedure. almost just a formality and not difficult at all (if the mother agree), no need for expensive lawyer

Good Luck

You say you are not married, so as far as the law is concerned the child is not yours and the mother has sole custody. As MikeyIdea said, to change that you have to legitimize the child.

You can also ligitimize the child according to your own countries laws, so the child might also have your nationality, etc. This is important and normally a country will recognize the legitimization according to Thai law. But a legitimization according to your country's law only will not be recognized by Thai law and there are countries that will not accept a legitimization according to Thai law only if you do it after the birth. Inquire at your embassy.

How do i go through the standard procedure at court without expensive lawyers?

It is just finding a lawyer that doesn't charge you too much to petition the court to recognize you as the father of the child and to be awarded joint custody with the mother.

Since the mother is agreeing it won't be difficult.

Cheapest way is to get the mother to call around for a cheap lawyer, doesn't have to be one speaking good English or anything. This is standard procedure - fact finding session and mother and father agree for father to be legitimized and get shared custody of child, you will never see a judge and it will take 5 months or so and judge signs everything, computers at amphur is updated - End

Cheapest I think should be around 40,000 bath for a lawyer in Bangkok (who doesn't speak English really, doesn't matter, he can read, use per and paper ) IF you trust the mother! 40,000, that's cheap in Bangkok, maybe 50,000 bath then. Make sure that layer understand that you want legitimization AND custody - parental rights. The Thai word used is Poo bock kraang = guardian, that's what you want to be

Good Luck