In short the mother had been separated yet never divorced, we had a son,the man left the country, I want to establish paternity, we are not together anymore, she is not sharing information on the ex, how can I get share custody?

You go to Court, ask for custody, submit to a DNA test.

How long ago was the child born and what State?

Two years ago, I'm in Florida

In all states, when a child is born to married parents the husband is considered the legal father until and unless challenged.

What's not clear is who you are. You may have no standing to challenge.

I'm the child's father, the mother and I lived together for thee years... her ex left the country five years ago but she never divorced

And you never went to court to establish paternity?