Phase 1. Godrick hits hard but slowly. Most of his axe swings and slams have a long wind-up, so the safest habit is to wait for the swing to start, roll through it, and take one or two hits before backing off. Do not get greedy after his five-hit axe combo, the last swings come later than you expect.
Watch for Stormcaller: he spins the axe and kicks up a cyclone that throws you back, then rolls into a wind-charged slash. Keep your distance while the wind is up and come back in once the follow-up lands. Storm Blade sends several homing wind blades at you; a late roll toward him usually slips through them. When he grinds the axe along the floor, an overhead chop is coming.
Phase 2. At around 60% health he stops, cuts off his own arm and grafts a dragon head in its place. From here on many attacks add fire, the dragon head breathes flame, and I Command Thee, Kneel! gets a third, bigger shockwave. Staying close to his legs and slightly to the side keeps you out of most of the fire breath.
If you talked to Nepheli Loux in Stormveil, her summon sign waits outside the boss fog. She is a solid tank for this fight. Spirit ashes are allowed too, and Roderika’s Spirit Jellyfish Ashes are a handy early pick for steady ranged chip damage.