If you are a J-1 visa holder planning to marry a U.S. green card holder (lawful permanent resident), you yourself may eventually be eligible for a U.S. green card. However, it's highly likely you will need to spend at least a few years outside the United States first, for the following three reasons:
Fortunately, you don't have to add up all the time delays described above into one grand total, but can wait them out simultaneously.
As the spouse of a U.S. green card holder, you are eligible for a green card -- but not until a visa number becomes available. In the meantime, you will be placed on a waiting list, based on your "Priority Date" -- the date your green-card holding spouse filed the required initial visa petition with U.S. Citizenship and Immigration Services (USCIS) on Form I-130.
The wait is often around five years. The only thing you can do to speed it up is if your spouse successfully applies for naturalization and becomes a U.S. citizen. In that case, you would become an "immediate relative," and could apply for your green card without further delay.
For more information on how to get a green card through marriage to a lawful permanent resident, see "Visas and Green Cards Based on Marriage to a U.S. Green Card Holder."
The J-1 visa was created in order to foster international exchanges of knowledge. For that reason, you may be expected to return home and share that knowledge once your visa expires -- particularly if your home country helped fund your U.S. stay. There's a good chance that your J-1 visa came with an automatic requirement that you return home, and stay there, for two years following your U.S. stay before attempting to return to the United States. Although waivers are available, you would have to have an immediate right to return -- such as a marriage to a U.S. citizen rather than a green card holder.
For more information on the restrictions on J-1 visa holders, see the "Exchange Visitor Visas" page of this website.
It is never advisable to remain in the United States beyond the permitted time on a visa. Once your permitted stay expires, your visa is automatically cancelled and you are in the U.S. unlawfully. Many people who overstay feel like they are getting away with something when no one from the immigration department shows up to deport them -- not realizing that the consequences usually come later.
For example, if you stay beyond your expected departure date by six months, and then leave the U.S. (as you'll most likely have to do), you will not be allowed to return to the U.S. for three years, despite your marriage. If your overstay lasts a year or more, you will not be allowed to return for ten years.
More information on these and other consequences of a visa overstay can be found on the "Visa Overstays: Rules and Consequences" page of this website.
Immigration law is always complicated, and your situation is particularly complicated. It is critical to hire an immigration attorney. The attorney will work to analyze the best strategy for you and your spouse, and help you to complete all the required paperwork.