In reply to Lh88: I believe that there is quite a lot of Alpine climbing done at that time of year (also in June), but it is pretty serious and needs a lot of understanding of conditions and as others have pointed out, mostly the ability to ski into and out of routes/huts. Having said that, I have seen quite effective use of snowshoes to approach if you are not confident of your ability to ski with heavy weights in serious positions.
North faces CAN be very good, but they can also vary from bullet hard ice to massive quantities of snow - there is not a general answer to your question, it depends on the area, the individual face and previous conditions. You also of course need to check the condition of descent routes in your area of interest, and to work out what you are going to do about climbing with skis/snowshoes or returning to your start point.
The weather can be glorious at that time of year, but the mountains feel much more serious, also as others have mentioned many huts will be shut, or rather unguardianed (roughly in the late May-June gap after the ski-touring season tends to stop and before the Summer season starts). Mostly you can check the opening dates of an individual hut on websites, they will invariably have a Winter room but then you need to carry food yourselves and sometimes stoves.
http://www.camptocamp.org/
is a good website about trips/conditions in specific areas, or you can ask about routes/areas here, there are quite a few people who live much of the year in mountain areas, though this will certainly give a bias toward Chamonix which is not always helpful.
So in summary, Alpine climbing at that time can be good to very good, but is mostly for the experienced and those well equiped to judge local conditions. If you have that ability, it can be very rewarding.