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.
That's a good way to do it, find what works for you and stick with it just be aware of the story arcs in DS9 and ENT so you don't accidentally jump into the middle of one, but even then that could be enough to peak someone's interest.
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.
Star Trek 6 and the TNG episode 'Yesterday's Enterprise' do a good job explaining this they are also fantastic to boot.