Godfrey's character is summed up by Eliot near the beginning of the novel as "irresolution and moral cowardice," a state in which he continues until almost the end. Godfrey is not evil in any way. He has no desire to harm anyone; he is pained when he does so. But he does not have the courage to take responsibility for his acts nor to give up his desires when they conflict with duty.
Eliot is careful to make excuses for Godfrey. The early marriage was not really his fault; he has good intentions about caring for Eppie; he really wants to lead a better life. Yet these are only the ultimate complication of Godfrey's situation. Such "excuses" are an insufficient defense. Godfrey knows that, and Eliot tries to make sure that the reader does too. She seems to be saying that there are times when wrong can be made to seem almost right, but that such conditions provide a weak base for a life.
Godfrey's life with Nancy sets him on a better path, but there is no true test of his character except his failure to own Eppie as his daughter. This shows that Godfrey is unchanged — he wants to do the right things, but not badly enough to risk his happiness. In the end, he tells the truth only because he is afraid it will be found out anyway. His own desires still are the most important thing to him. He puts them in the form of principle now — he thinks he has a "right" to his daughter, although he was willing to neglect the corresponding duty.
Godfrey at last comes to some self-realization. The unexpected resistance he meets from Silas and Eppie brings home to him for the first time the fact that rights and duties cannot be separated. He accepts his rebuke willingly. Still, he fails to do his whole duty. He takes the easy way out, deciding to "own" Eppie only in his will. But at least he does it from better motives, from consideration for others rather than for himself.