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Don't fall into the trap of assuming that you have to act out how dialogue is spoken. Yes, some author do try indicate every nuance of sound in some character's speech, but most readers find it highly annoying. 

Fortunately you don't have to rely on acting it out. Your book has a narrator. You  narrator can simply say that the character had an incomprehensible accent the first time they speak. 

To remind the reader that this character is hard to understand, you can then throw in just a few hints of their accent. Not enough to be illegible, but enough that we remember that this character speaks this way. So you might do something like this:

"I'm gowink down to the river."

Bad example, but by indicating the sound of just one word here and there you can remind the reader without confusing and them. 

Also, remember that people who speak very differently from those around them probably have more differences than just their accent. Accents will normalize with contact, so there has to be some cultural and experiential differences as well. Think about what would this person say that is different, not simply how differently they would pronounce the same words.