“For David says concerning him, I beheld the Lord always before my face. For he is on my right hand, that I should not be moved. Therefore my heart was glad, and my tongue rejoiced. Moreover my flesh also will dwell in hope, because you will not leave my soul to Hades, nor will you give your Holy One to see corruption. You made known to me the ways of life. You will make me full of gladness with your countenance.”
These words are based on Psalms 16:8-11 LXX being almost word for word apart from the omission of ‘at your right hand there are delights for ever' in LXX. While those words would have made the case stronger Peter feels them unnecessary for his case. Note the expression of total loyalty to God in the Psalm, without which what followed would not be true, the confidence that as God's ‘holy one' (i.e. as His anointed who is faithful to Him) he will not be left in the grave or be allowed to suffer corruption. Note also the certainty that he will again experience life and be joyful before the face of God. Whether the writer of the Psalm was originally here expressing his hope of a future life, or was simply expressing the hope that God would not leave him to an early death in the situation in which he found himself, is disputed, but the words not quoted by Peter support the case that he was thinking of living for ever because he could not believe that God would forsake him or let him sink into oblivion. This idea appears in a number of Psalms (see Psalms 17:15; Psalms 23:6; etc) and Isaiah too would cry, ‘My dead bodies shall rise -- the earth shall cast forth the shades' (Isaiah 26:19) in a context speaking of Sheol (Hades - the grave world - and compare here Psalms 139:8-9). But the distinction is of secondary importance here because Peter goes on to explain his argument.
It is unlikely that we are to see Peter here as specifically using Rabbinic methods of exegesis. It must be seriously doubted whether he knew of such methods as such. What he was doing was using methods that he had learned from Jesus, and which were generally recognised by the common people from their contact with Pharisaic teaching, combined with good common sense and spiritual insight, fortified by the Holy Spirit.