I can only speak of the North Pacific. It may well be different on the North Atlantic.
If you need to deviate for weather you should always call ATC for clearance to do so.
If you need to deviate but CANNOT cannot contact ATC for clearance then you can deviate up to 10 miles left or right of track and still maintain your level. You must of course switch on all lights and infrom other traffic on 121.5 and the chat frequency of of your action. TCAS should also be carefully monitored.
If you need to deviate more than 10 miles but CANNOT conatact ATC for clearance to do so you should maintain your level until 10 miles from track and then climb / descend 300feet when greater than 10 miles from track. The requirement to climb or descend depends on your track and whether you are deviating left or right of track.