The Russian word for devil is чёрт. It's a bit irregular because it becomes soft in the plural, thus it declines like this:

In English it sounds rather quaint to say "Go to the devil," but in Russian the phrase is much stronger and not quaint at all. To catch the emotional color of it in English, it is best to translated it as "Go to hell":

Another way to say about the same thing is to literally say "The devil with you":

In the US many Christians consider it offensive to take the Lord's name in vain. In Russia it is more offensive to swear using the word "devil." Interestingly enough, by saying "to God with him" they can express indifference or minor irritation. It's not nearly as strong as saying "The hell with him" and can be said in polite company, so I think it is best to translate those phrases into English with "To heck with him/her":

Of course, if you are a sensible foreigner, you aren't going to go around Russia telling the Russians to go to hell, but you will still encounter the word in real life, art, and literature. My favorite "to hell with you" is in a song by Веня Д’ркин called «Ты нравишься мне» “I like you”. Ah, rage and betrayal and desperate love all wrapped up in one melodious package. You can listen to the song here, along with a translation.