In reply to SteveD: What sort of routes are you after? Also, can you ski if there is heavy snow?
Late June can be excellent, with good settled weather, while crevasses are normally filled and easy to cross. On the other hand, if you are doing "foot Alpinism", there can be lots of deep, unconsolidated snow at that time of year which can involve paralysing wallowing. It very much depends on how snowy the Winter/Spring has been.
It is not normally too crowded at the end of June as it is still before peak season, but the check that the huts you want to use will be open.
Late June can be very good for Alpinism.