If you marry a someone that entered the country illegally and was then deported back to their country, how do you dissolve the marriage?

Is the divorce amicable and uncontested or is your spouse ging to contest the divorce?  Do you know where your spouse is now?  The key to your divorce is notice to your spouse of the legal proceedings filed against them.  The Court must have the power to decide a case against the parties to the proceeding.  This is known as "jurisdiction."  The court obtains it in the first instance by filing in the correct court and through the plaintiff - that would be you.  Each state has a residency requirement for bringing an action for divorce.  You must meet it to file along with proper grounds for divorce.  You must also serve the other party with a copy of the divorce papers. Many states allow the defendant - your spouse - to file an affidavit if they do not intend to appear and answer the petition filed or to "waive service" allowing the court to proceed.  Seek legal consultation from an attorney in your area as to your particular set of facts.  Good luck.