It depends on your perspective. They would be unmarried if the marriage wasn't valid for some unforeseen reason. If they married knowing they were already married they would be a bigamist. They would be a victim of fraud if they married a bigamist unaware that he/she was already married.
If you are asking for the term, bigamist is the word.
this person is called a bigamist
A hacker tries to access a computer illegally.
You would call that forgery on a black market and the person selling it a con artist.
Dealer, supplier or distributor...
There is no way to get illegally married and then acquire any of the legal rights of marriage.
The Bridegroom.
A person that is about to get married is called a bride (female) and a bridegroom(male)
If you are here illegally, he can. If you are here illegally, he can.
a female who isn't married is Mademoiselle
There is no such thing as illegally married gay couples. There are only legally married gay couples (with or without state recognition), and unmarried gay couples.
...is illegal I can only do so much with incomplete sentences.
paramour
an individual is illegally married