Both Psalm 69 and Psalm 109 are psalms of David. Both are, as well, psalms of vindication. In Psalm 69, David's enemies seek to destroy him without cause (v4), so David prays out against them that God would vindicate him against them. Similarly in Psalm 109, David has been betrayed by his friend(s). Therefore, David prays that God would destroy his enemies.
Peter, as you mentioned, picks up these two passages in Acts 1 to demonstrate the need to replace Judas among the twelve. There are two main theories that I am aware of in regards to Peter's use of these passages:
- Peter has no real hermeneutic. He's simply proof texting.
- Peter sees these Psalms through a Davidic typology.
There is some evidence that Peter's categories allowed for typology. Although it wouldn't be written until after the story in Acts 1, in 1 Peter 3:21, Peter uses the word "αντιτυπον" (meaning something like "antitype") to relate the deluge to baptism.
Moreover, the gospel writers all seem to think that David in some way prefigures the Christ.1 This is not too surprising, though, since the prophets also anticipate a king like David.2 This lends the second theory a bit more credence since Mark is thought by many to have based his gospel on the testimony of Peter - though again, the gospel would have been written undoubtedly at a later date from the story in Acts 1.
It's a least possible, though, that Peter has in mind a Davidic typology when he quotes these two verses. If he does, then he is looking at the sufferings of David at the hand of his betrayers and the prayers David prayed and seeing in these things a pattern of the David-like Messiah and his betrayer. It then becomes appropriate of Jesus' betrayer - like David's betrayer in the psalms - that his "camp become desolate" and "another take his office."
1 See Mark 11:10, 12:35-37, and 15:24 (cf. Psalm 22:18)
2 See e.g. Jeremiah 30:8-9 or Ezekiel 34:23-24