To give the best answer I think you might have to clarify exactly what you mean by "crimes by the government." Like you, INAL, but I do have some understanding of various aspects of the law for various reasons. To really be sure you would probably want to speak with a lawyer that practices in this particular area, especially since there are some unusual aspects to it compared to ordinary criminal law or the law of war. Having said that.....
The US government has what is known as sovereign immunity. It has to agree to face legal consequences for its actions in court for anyone or any organization to take legal action against it in US courts. There are many areas in which it has done so, and others where it hasn't. When you say "crimes by the government that cross national boundaries," I'm going to assume you are referring to intelligence gathering or surveillance. US Law and constitutional rights, as I understand them, are largely confined to American territory, or vessels, although American citizens retain their rights outside the country when dealing with the US government. A citizen of Syrian living in Luxembourg as a member of a terrorist cell plotting attacks against Canada has no rights under the US 4th Amendment that would require the NSA or CIA to get a warrant to spy on him. The same would apply to the Quds Force special forces of the government of Iran. The NSA or CIA wouldn't require a warrant to spy on them. The same would apply to other countries and their citizens. Inside the US, the rules change so there would need to be warrants at some point, unless they were in direct contact with terrorists outside the country. (And you can quibble about this point on various statutory or Constitutional grounds.) And American citizen would retain 4th Amendment rights both in and out of the country unless they were in direct contact with a terrorist group. (Same quibbling could again apply.) So if some US intelligence agency actually did have access to an email account of a foreign leader, it is very unlikely that there was a crime committed under US law for there to be an action against the Federal government in US court, even assuming that the US government waived sovereign immunity in that instance, which isn't likely as far as I know. (Check with a lawyer.) There might be a diplomatic problem, but that is a different question. If some foreign citizen felt that they had a legitimate grievance against the US government, the thing to do would be to contact a lawyer that practices in the area of US law in question and see about filing a suit US Federal court. It would start in the lower courts. If there was a significant Constitutional question, it might make it to higher courts, perhaps even the Supreme Court. I think the key for the vast, vast majority of people to avoid being a subject of surveillance by the US government, when someone is actually looking at your information instead of just having it in a computer, is to avoid involvement with violent extremists groups or foreign intelligence agencies. The resources that the US has, extensive as they may seem, are still limited and they aren't going to waste much time on someone unless necessary. Write letters of protest instead of picking up an AK.