I'm doing a similar thing right now, and I am watching them in order - however I'm aware this may not work. My plan is if I feel it becoming a chore, I'll just go off and watch an episode of, say, VOY, or DS9, and then come back to TOS, where I currently am.
I decided to do it this way because all five series share fairly good continuity. I realised that by just jumping in with TNG, I had no idea how the Klingons and Federation came to be on good terms.