With Nepheli's help you can effortlessly stagger Godrick and put a decisive attack in. You should let her be the focus, then get in Elden Ring Runes many forward and side strikes as Godrick is trying to strike her. Make sure to pay attention to his movements because he can return his attention to you at any moment.
With his hand of the dragon on there are only a handful of other moves worth paying attention to. Most dangerous is the dragon breath, which covers huge effect that can be sprayed multiple times. He telegraphs this one by raising the dragon head into the air as its mouth begins to release flames. Move it to the side or backwards so that you don't feel the full force of the blast. The other option is the fiery rock strike that communicates by dragging the dragon's head across the ground in a circle before blasting the rocks with a fiery blast into the air. Then, you can sprint toward him and let the rocks sail on top of your head. Or simply go away and recover while the rocks burn on the ground.
If you feel more confident , you can get an Gold-Pickled Fowl Foot and use it prior to killing Godrick to bring in additional Runes. In buy elden ring weapons the end, if you defeat him, you will receive the Godrick's Grand Rune and The Remembrance of the Grafted (See our guide for ways to replicate Remembrances).