It is, as has been mentioned above, a practical impossibility to ride round close to the whole circuit - the best you can hope for is to cycle between the various vantage points. Basically you can see the following sections legitimately and I've included a couple of other bits that you might be able to blag your way into !!
Main Start finish area - basically from the end of the Porsche curves into the chicanes all the way round to Tetre Rouge is readily accessible and there are plenty of really good spectating points. I believe you can also get quite a good view from the cafe on the outside of Tetre Rouge, but I've not tried it personally.
Mulsanne straight - one official access route which is to the restaurants just before the first chicane. Best view is from the tables in the Chinese restaurant !!
Mulsanne straight - there is a hotel / auberge down near the kink (between 2nd chicane and Mulsanne Corner) which you can sometimes gain access to. The owner was a bit stroppy in recent years and asked us to leave even though we just wanted to sit and have a beer. If you're not booked into the hotel or possibly just for dinner then expect to be disappointed.
Mulsanne Corner - good view of the cars braking for the corner (from the banking) and the spectator enclosure extends along the track towards Indianapolis
Indianapolis / Arnage - another proper spectator area. Easy enough to access but expect traffic on the one way system on race day
Porsche curves - the new(ish) enclose where you can see the cars on the run up from Arnage and turning right off the road section of the track into the Porsche curves. I've also seen people spectating from the grassy bank on the *inside* of the corner (opposite the formal spectator area) but that may just have been people associated with the marshalls.
Pretty much everywhere else is off limits and patrolled by marshalls / gendarmes.
The flow of traffic on the roads around the track is controlled when the race is on by virtue of some well sign-posted routes. Basically the one-way system flows in the opposite direction to the way the track runs. It's deady easy to get around, but can be a long ride if you're on a bike !! It's a BIIIIGGGG circuit !!
I'm sure there are other areas people will say you can get to, but these are the "safe" ones we found.
Bit gutted I'm not going this year. Still, 2004 is probably on the cards !!