As Shakespeare was writing under the rule of a queen who never truly believed in either forms of monarchy, in this play he appears to have a free hand in deciding which one to more strongly portray. And after the first two scenes it appears that he wants to be even handed between the two.
In Act 1 Scene I both Bullingbrook and Mowbray start of their appearance in the court of chivalry by being very and perhaps overly respectful to the king. Bullingbrook pays tribute to the king as a feudal monarch, "my gracious sovereign, my most loving liege," (1:i:21) where as Mowbray seems to think of the king as divine saying, "until the heavens, envying earth's good hap, add an immortal title to your crown" (1:i:23). So right from the very start Shakespeare seems to be playing with these two ideas, he does not chose between them but lets the characters speak of both of them and let us decide. This respect to the king continues for the main speeches of the trial with Mowbray saying, "the fair reverence of your highness curbs me" (1:i:54). But whilst this recognition of the king's status is interesting it seems all to obvious that the two men would pay respect to the most powerful man in England; and so, I believe that Shakespeare cleverly introduces these ideas as part of the formalities whilst making them important features of the play.
After the formalities Shakespeare introduces how the monarch might feel about kingship. In 1:i:115, Richard says, "impartial are our eyes and ears", but how can a feudal monarch be impartial as they need the support of their barons and favours from the nobility, so surely Richard does not consider himself feudal and impartial? Well it appears not as shortly after he says, "such neighbour nearness to our sacred blood" (1:i:119), sacred blood is a clear reference to him being appointed by God and so it appears that Richard believes he was divinely appointed.
