Just for the fun of it I made a list of Psalms which seem to clearly express a close personal relationship with Jesus (setting to one side but not forgetting that they are frequently addressed by the Son to the Father, for example 2:7 "The LORD said to me, 'You are my Son, today I have begotten you'" or 3:3 "But you, O LORD, are a shield about me, my glory and the lifter of my head"). I deliberately excluded any Psalms that didn't explicitly use "I-you" language eg Psalm 1, Psalm 11, or any that seemed borderline eg Psalm 12; arguably this was a mistake.
I found these fit the pattern:
2 3 4 5 6 7 9 10 13 16 17 18 ("I love you, O LORD, my strength") 22 23 25 26 27 28 30 31 32 34 35 38 39 40 41 [42 is excluded only if you do not link it to 43] 44 51 52 56 ("You have kept count of my tossings, put my tears in your bottle.") 57 61 62 63 65 66 69 70 71 73 77 84 86 88 (by speaking of its opposite) 91 (twisted by Satan because he implied that the relationship might *not* be that close) 92 94 101 102 103 ("who redeems your life from the pit, who crowns you with steadfast love and mercy, who satisfies you with good so that your youth is renewed like the eagles." Not 'I-you' but surely deserves inclusion) 109 116 118 119 121 123 130 138 139 ("O LORD, you have searched me and known me!...Even before a word is on my tongue, behold, O LORD, you know it altogether.") 140 141 142 143 145.
It could reasonably be argued that this exercise is of limited value on these grounds:
(1) Every Psalm, even where a personal relationship with God is not spoken of explicitly, assumes the reality or the potential for such relationship. Therefore, the list should include all 150 Psalms.
(2) Every Psalm is intra-Trinitarian in nature, expressing the relationship of the Son to the Father through the Spirit. Therefore, it is not possible for them not to be deeply personal in character, and when we use them for our own, we are expressing this personal relationship 'en Christo', in Christ. Therefore, the list should include all 150 Psalms.
(3) If we note that many of the Psalms are 'of David', then our relationship with God is closer than his, because we are of the New Covenant, and it is impossible to imagine a negative answer to the prayer "take not your Holy Spirit from me" (Psalm 51:12), for this would involve our Lord in teaching his children to pray a prayer that he then refused to honour, which would contradict his promise to us in Luke 11:10-13.