But it isn't French. What is it exactly: Old French or Latin?
Michael Hirst, creator and writer of Vikings, told this to Medievalists.net:
And here’s a very important thing: if you include Latin as a dead language, we have FOUR dead languages in the show. We have people speaking Anglo-Saxon, Old Norse, Old French (when they raid Paris, which sounds more like German than French, actually) and Latin, and can you tell me ANY other network show that has people speaking four dead languages in it? No! I mean, they don’t speak it much, but you can tell that is what they are speaking. I wanted people to hear what those languages really sounded like. It’s a very important part of the authenticity of the show.
I know this is old but I just wanted to add to the above answer where it says the French sounds more like German, I believe (despite the words of the creator since he could have easily mixed this up in speaking) they are speaking Old Frankish rather than Old French, which is actually a Germanic language so that's why it sounds German and it makes more sense for the time period.
As a francophone, I can catch many of the words that the Parisians speak. Of course, the subtitles help, because the accent and grammar and morphology is very different. But, it is believably the ancestor of the French language.