This is an interesting and difficult question. Unfortunately, not much is known of Viking equipment, including clothing, because such military goods were relatively expensive and rare. For example, in those times (800-1000 AD) it was common for people to go barefoot, shoes were so expensive. Written works from the time rarely discuss viking clothing in any great detail. Also, it is important to remember that clothing would be vastly different from one person and one area to another, the "vikings" having many different sub cultures. For example, what a Swedish viking from the lakes region might wear would likely be very different from that worn by one from Trondheim in Norway.
The most authoritative book on the subject is Paul Norlund's "Klaededragt i Oldtid og Middelalder" (1942) which unfortunately has never been translated into English.
From many different miscellaneous writings we can infer as a general rule the basic clothing was three layers: a linen "kirtle" or undergarment, a wool coat, and then a chainmail hauberk. Note that a "coat" at that time meant a long-sleeved garment that went down to the knees and usually had a hood. In addition to this well off soldiers would have a helmet and boots.
In addition to this basic pattern it is clear that many wore animal skins in one form or another, and this would especially be true in cold weather scenarios. For example, fur boots, fur mittens, fur hats and fur cloaks were all common articles of trade both among civilians and warriors. Note that by using a tallow or wax treatment hides can be made relatively waterproof.
Seafarers also probably used sealskin garments which were very expensive and valuable. Sealskin is waterproof and was used both for clothing and for dressing small boats. Sealskin can be made into caps, mittens, booties, tunics and other specialized gear for wet weather operations. Such valuable gear would be carefully removed and stowed when landfall was made.