Not much first hand knowledge on north atlantic procedures, i always flew in the VHF region which basically worked like normal flying anyway, except the required oceanic clearance which we usually got via datalink.
That said, in cruise there have to be regular time and fuel checks. Usually done on the OFP (operational flight plan) or the responding app, at least once an hour, but some companies require that to be done every 30 minutes or each waypoint, whichever comes first. In that we check how we are in time (late, early at a waypoint) and fuel (burned more, or less) and therefore how are we doing in regards to arrival time and remaining fuel and possibly extra fuel, contingency fuel etc. And of course we regularly check weather at each possible en-route alternate as well as our destination and alternates for that. And obviously the usual stuff about in flight weather avoidance is always ongoing (turbulence, convective weather).