It's too bad that you are going in June - the inland roads are usually still closed then. I spent a couple of weeks in Iceland in early August 2010, I drove around the country. As much as possible I stayed away from Highway 1 (Ring Road), preferring instead to drive the two-digit roads or, better, the three-digit ones. My route was counter-clockwise from Reykjavik up to the W Fjords over to Akureyri down to Egilstadur to Hofn and around through to Vik and back to Reyjkavik. I also spent a day in Landmannalaugar in the Interior of the country. That day was a highlight of the trip.
There are few to no trees in Iceland (except near Egilstadur's national forest... even then they are sparse and not very tall, think Seymour Demo Forest km 8 on Mainline) so camping will be on open areas. The weather changes rapidly. And there are microclimates. You will find yourself driving in blue-sky weather, over a pass, into thick fog. Or rain.
The country has 3G throughout its perimeter, if you bring a smart phone make sure you bookmark the weather map. It indicates weather and road conditions - indispensable on a trip such as this. I scoffed when others told me about the changing weather, then got into an incredible ocean storm on the road to Hofn, with mountains on one side, and stormy sea on the other. I've never seen anything like this in BC or anywhere else I have traveled.
The country is beautiful, the people friendly. Most speak English. The only people I met that did not speak English were in a remote town in the W Fjords, but we made do anyway. There are gas stations at frequent intervals. The price then was spendy at 1.65 cents a litre, but not insanely so. Food prices too were Whole Foods-grade; again, not the insanity of pre-2008.
I loved the North of the country. The empty fjords upon working my way out of Isafjordur. No traffic behind me, no one coming towards me - at the height of tourist season, even. The little homesteads, spaced out from one another, kilometers apart. The ubiquitous little churches, nestled at the mouths of the fjords. The immensity of solitude, just me and my car traveling three-digit gravel side roads, blind curves, hairpin turns, no one else on the roads. Five bars on the iPhone though. And illuminated signs every few dozen kilometers, indicating the wind speed and temperature at the next pass.
Places I remember with fondness - the emptiness of the W Fjords. The mystery of Asbyrgi. The roiling sands and hissing rocks near Myvatn and the geothermal station nearby. The beautiful waterfalls. The forest at Egilstadur. The storm near Hofn. The icy marvels of Jokulsarlon. The peaceful green fields of Vik. Oh I still dream about Vik. And the surprise of the Blue Lagoon (which I resisted going to) I spent the better part of the day immersed there, talking with people from my own countries and others far-flung.
Iceland is quirky, expensive, and will use up all your photo media cards. Yet I returned home and regretted not taking more pictures than I already had. The rocks, the mosses, the landscapes, the geysers, the rivers, the waterfalls, the icebergs, the vast stretches of open empty roads where not even dogs or sheep dare roam... perfect escape, perfect reset button.
If you haven't seen Heima by Sigur Ros, view it. It presents a good travelogue of what you may see on your way around. But it's only a small part of the picture. Take your cameras, make sure you take your rechargers (european plug system, take adapters) and click away.
And whatever you do, don't miss the little puffins at Latrabjarg. They are delightful and absolutely not shy.