Firstly, I haven't cycled this route but I know much of it from cars and motorbikes.
I think that there is a Sustrans route (I can't find which on at the moment, I find the route guidance bit of their website appalling but I think that it's National Route 5) which will get you from Anglesey to Reading. From Reading, you can go via Basingstoke, Alton, West Meon then Waterlooville then drop down to Portsmouth.
Good luck, it's a long ride.