In reply to musashi:
This booklet is pretty useful:
http://www.vegagerdin.is/media/umferd-og-faerd/Opnun_fjallvega_en_2014.pdf
There could be an early spring, in which case you'll be fine, but you could find yourself unable to get anywhere because the roads are closed. I was last there in summer in 2013 and the F225/208 didn't open until late June and the northern section of the F26 until the first week in July. This was a late summer, but it gives an idea of what's possible.
On the Laugavegur you'll be fine by yourself if you're a reasonably experienced backpacker but I'd gain experience of Iceland before heading away from well-marked routes. Tackle river crossings with caution and beware maps: they will often be inaccurate and will show paths where there's nothing on the ground whatsoever.
Chris's (oversaturated) photos show what the weather could be like, but on our long trek in July 2013, I was wearing Scottish winter clothing most of the time. Expect clag and you won't be disappointed!