Haven't been myself but I know a fair bit about this particular business.
Connecting the actual locations of all those "big name" climbs in the French Alps requires a lot of research in order to "triangulate" a place to stay, so I can help you with something I created via my own website.
Try this link:
"Legends of Cycling" via Google Maps
There isn't really a single place to stay that covers everywhere.
Bourg d'Oisans is popular and is also right at the foot of Alpe d'Huez (no through road at the top of that, so no value in staying up there). You can do the Galibier/Lautaret from there with a good, flattish run-up.
Elsewhere, I'd probably look at St Jean de Maurienne, which is on this year's TdF route, and Briançon. I've even heard of people staying in Sestrière. There will also be those niche out-of-season ski resorts, but cheap they ain't.
=========
Other stuff:
=========
Quite a lot of high passes (> 2000m) are usually closed right up to the end of May. You should be more than OK by June though, even after this hardest of winters, provided that the weather's generally friendly.
In June it can sometimes be still a bit unpredictable with the risk of heavy fogging higher up, which means less panoramic views and taking extra care on the descents!
Assuming you're more or less sticking with France, keep up to date with the situation nearer the time on the Bison Futé (The Wise Bison, don't ask
) website. The page you want there is entitled
Accès aux Cols. Note that this particular section is still only available in FR but it's straightforward enough to use.
General cycling-related blurbs here:
ClimbByBike (very good coverage)
Grenoble Cycling
Bags of info can be gleaned free from Google Earth. Elsewhere
IGN.fr sells excellent hard copy maps and you can, for a bit of cash, customise your own. Many are available in small formats for the jersey pockets. In the UK, your best bet is
Stanfords, which also sells IGN maps and guidebooks. Of course, you might be using GPS, so try
wikiloc.com for ideas if not actual routes.