It seems that you are asking about your marriage to an undocumented immigrant (a/k/a an illegal alien). The lack of a legal Immigration status has no effect on the validity of a marriage. Persons illegally present in the United States can marry citizens, or marry other undocumented immigrants, and their marriages are perfectly legal.
If you married a foreigner (whether he has a visa, a green care or neither) then you remain legally married to him (even if he returns to his country) and you may not marry again unless you first obtain a legal divorce.
divorce
I'd get a lawyer as fast as I could.
You divorce and get custody over your kids or just get full custody!
You need a divorce!
She can't if she wants to stay in this country.
In NYS you can file for a no fault divorce without getting consent from your husband. As a result, you will not need his information to serve your husband divorce papers.
That depends on the law of the land in which the marriage took place.
Yes, you can still get a divorce. You need to contact your husband so the correct papers can be signed from your lawyer.
Each jurisdiction has its own set of grounds for divorce. You need to check your particular one.
Witnesses are not required. A divorce is a private matter between the husband and the wife.
If the immigrant married a US citizen, then that's a done deal and she is no longer illegal. It sounds like you need to take care of your own affairs, starting with divorce proceedings.
It is quite simple. You need to tell your husband you want a divorce, and then talk to a lawyer.