Weather will be better around June, but it may be possible to dive all year round.
The islands are different, so to chose an island you should say more about what kind of dives you prefer.
El Hierro is probably the one with the best life and some nice underwater landscapes. I'd recommend
Benthos Buceo El Hierro - Página de inicio Website in Spanish, but go to "Contacto" and give them a shout.
I have also enjoyed Lanzarote. Mostly shore dives there. Try to dive both the south and north sides of the island.
In Gran Canaria there are some big wrecks near its capital. But it's not the touristic area. When I dived there I was the only tourist on the boat, all others were locals. The south is more touristic, did shore dives there and there's nice life.
La Palma has some interesting spots. Mostly shore dives as well although a centre sometimes has a boat in operation.
Tenerife is ok, but not the best among these islands, although I don't remember much and only did 3 dives. I remember barren landscapes...