How would it be possible for someone to marry an illegal alien from Mexico who has been in trouble with the law and for this person to still be able to become legal and how long would it take?

If an illegal has been arrested for a felony ANYWHERE in the United States, he can be arrested and put in prison for 1 to 10 years for aggravated re-entry. After he serves his prison term he will be immediately deported and any time he is subsequently picked up even for spitting on the sidewalk, he will be put in prison again and subsequently deported. We already have far too many illegal aliens in this country. Don't encourage him.