I went for the first time back in April last year, and I've a stay scheduled again for the end of June. So not a seasoned expert but:
They should send you a list of required stuff such as bedsheets and what not. In April (and with the late spring we had) I found the building quite cold and damp. June/July will hopefully be better, but I still recommend packing your fresh clothes wrapped inside plastic bags. There is no shop on Skomer, so you'll need to take everything (and extra. You can get stranded for extra days if the weather turns!). You also have to take everything home at the end (minus organic waste that can go on the compost).
There are two electric sockets in the kitchen. When I was there I was one of only two photographers, so I pretty much free to recharge when I wanted (a couple of phones were the only other things I saw them being used for). I suspect in June/July there might be a bit more of a queue so plan on the assumption they'll be limited.
I also had the puffins to myself for long periods, which isn't going to be the case in the peak season of July. I heard it can sometimes get a bit scrum-like when it's at the most busy. The day trippers (potentially 250 of them) start to arrive at 10am and all rush off to the other side of the island for the Puffins. But they'll off by 3-4pm. In July, that gives you many hours either side when there's going to be a maximum of 16 guests to share with (plus the staff and volunteers). Use the middle of the day to shoot other stuff, have a siesta, shower, etc.
When photographing the puffins, keep an eye out for any that want to pass but are being timid. With so much going on, it's easy to become focused on the ones in front of your lens and not notice others around you. Be prepared to get up and move out of the way.
Gear wise: I took better photos with my 70-200 f2.8 than my 100-400. I've heard people say "You don't need long lenses, the puffins are that close." However, pros still turn up with 500mm+ lenses, and they have their usefulness. (I'd take one if I could afford one!).
Every night there is a get together when they record what everybody has seen and have a chat. I found there was a great communal spirit between the trust staff, weekly volunteers and the paying guests. With sunset times as they are in the summer, I might miss more of these this time around, but if they don't clash with good light, I'd definitely try to attend.
If you want a feel for the island before you go, there is a decent book by an ex-warden
http://www.amazon.co.uk/Skomer-Portrait-Island-Jane-Matthews/dp/1905582587
Also: don't drop your camera over the side in to the sea (happened a few days after I was there so somebody on a photo workshop trip)