Hi Meles,
In 1998 I worked as a guide one summer on the RV Professor Molchanov. The cruises were organised by Oceanwide Expeditions.
Usually the ships start in mid June and the season ends in end August. For seeing most of the birds and mammals I would recommend an earlier start, certainly not August. June or July, well this depends on the ice which is very unpredictable.
We had polar bears on all trips, although it's more difficult to find them when the ice is gone. Most (14 on one trip of 10 days) we had early July.
Walrusses were easier later in the summer. On the first trip in June we had a group passing by. When the ice is gone and the ships can approach the island of Moffen (landing not allowed) they are pretty guaranteed. As long as there is still some ice the chances of finding one or two on an ice shelf are also pretty good.
Auks (Brünnich's Guillemot, Little Auk) are also easy to see, especially when the Hinlopen street is ice free (usually early July to mid July). In August they are finished with breeding and you will find them at sea.
You see, it's not so easy, as it depends very much on the ice which changes every year. Still I would recommend you early or mid July.
Try to get on a smaller ship (maximum 50 passengers or so) as your chances to see something are much better on them. Also check if they are doing zodiac cruises and walks as some birds or mammals are much better to see then.
I certainly can recommend my former company (
http://www.oceanwide-expeditions.com/
but there are also other good ones. If you need more precise information or if you have a question just pm me.
Enjoy your arctic experience then
André