What is the immigration process for an expatriate spouse married to a US green card holder?

You have to file I-130 and when your petition for your spouse is approved, your spouse will be notified by the Department of State when a visa number becomes available. The process take approximately 5-6 years, if you become US citizen after filing the petition, you may upgrade to "spouse of US citizen" which takes 8-10 months.

If your spouse is outside of the United States at the time of notification, he/she must then go to the local U.S. consulate to complete visa processing. If your spouse is inside US through a lawful admission or parole and is maintaining that status at the time of notification, he/she may file the Form I-485 when the visa number becomes available.

If you do not have the visa number issued by the Department of State, you must wait for a number to become current. Your spouse may need to depart the United States to avoid accruing unlawful presence.

If she currently is in US illegally, you will need an experienced immigration lawyer as your spouse will be barred 3-10 years depending on length of overstay.