No, Griffin did not follow the saying 'with great power comes great responsibility'. The saying means that if one has the ability to do something, one should make sure to do it for the good of others. Since Griffin did not think of anyone else except himself, this saying is not what he followed.
After he managed to make himself invisible, he did not use his powers of invisibility for doing good. Instead, one of the first things he did was to set fire to his landlord's house in an act of revenge. He did not think of helping anyone with his newfound powers, instead, he harmed other people for his own gain.
He stole clothes and food for himself without feeling any remorse. He even attacked and robbed a shopkeeper. When his money ran low once more, he again resorted to stealing and robbed a clergyman. When Mrs Hall entered his room, he attacked her with her own chair. Finally, when the police were about to catch him, instead of going quietly and accepting his arrest, he fought back and escaped the police. At no time did Griffin take any responsibility for his actions.