I take my Landcruiser to Iceland every year, this August I am trying to get a heavy duty rack to fit on the rear crossmember so I can carry my crf 230 as the tracks are quite spectacular, especially when going inland and up to the north west which is virtually uninhabited.
Going it alone is a definite no no for the obvious reasons.
Actual 'off roading' is not permitted although in the winter when it's all covered in snow you can go wherever you want, the locals have a blast in the monster trucks over there.
The permitted routes are graded for difficulty, see attached image. Some of the more extreme routes are quite a challenge for bikes, believe me. Some of them go on for up to 60 kms without a break. Very varied terrain, steep hills, rocks and lava sand not to mention the rivers
Personally, I would not go there with just the bike unless the plan was to stay more or less around 'civilised' areas. Reason being once out in wilderness you are at the mercy of the weather, blizzards can come in August and the rivers are glacial so if camping you could end up having a bad time to say the least.
It is a fantastic place, often described as Scotland on steroids. I used to go to Morocco but this place has put paid to that.