How does Western Christian theology explain the doctrine of the Filioque within the context of the Old Testament? I'm doing a pseudo-superficial scriptural study of the Nicene-Constantinopalitan Creed and how it differs from the Latin addition of the Filioque. The Caroligian theologians mainly used the Gospel of John's words of Jesus where he says that he will "send" the Holy Spirit from the Father.

I'm having trouble wrapping my mind around how this applies to the co-eternal relationship between the persons of the Trinity. How does the Spirit "proceed" from the Son before he is sent by Jesus, given that scripture indicates the Spirits procession from the Father before the incarnation?

Ref. Psalms 51:13, Isaiah 63:10
If I understand the question, it seems to be along the lines of "How can Jesus be the one who sends the Holy Spirit, when the Holy Spirit is sent in the Old Testament?"

There's two threads that help answer this.  The first is that Jesus is there in the Old Testament.  The second is that the Holy Spirit was only given to a select few in the Old Testament, not all the people.

Regarding the first thread or point, Jesus is the incarnate Word of God.  He is the same Word through whom God spoke everything into existence in Genesis 1; John makes this point in the opening of his Gospel.  Jesus is also the "angel of the Lord" who appears throughout the Old Testament.  The "angel of the Lord" is the Lord Himself, the pre-incarnate Jesus Christ; this is seen, for instance, in Genesis 22, Exodus 3, and the visit of the three men to Abraham in Genesis 18, one of whom Abraham calls "the Lord." So, Jesus was there in the first act of creation and is with his people throughout the Old Testament.

Regarding the second thread or point, in the Old Testament the Holy Spirit is only given to certain people, mainly the prophets (although the kings such as Saul, David, and Solomon also had the Holy Spirit, see for instance 1 Samuel 16:14 when the Spirit departs from Saul).  In fact, in Numbers 11:29, after the Lord takes some of the Spirit He had given Moses and gives the Spirit to the elders of Israel, Moses says, "Would that all the LORD's people were prophets, that the LORD would put his Spirit on them!"  Then, in Joel 2:29, the Lord promises that He will one day pour out His Spirit upon "al flesh."  Peter connects this promise with Pentecost in Acts 2:17-18; this is, in the New Testament Church the Lord is fulfilling His promise to pour out His Spirit upon "all flesh."  

Thus, owing to Christ's ascension back into heaven to take his seat at the right hand of the Father, the Holy Spirit is poured out on all believers, not just the select few as in the Old Testament.  This is visually seen in Revelation 5 when John sees Christ, the Lamb "with seven horns and with seven eyes, which are the seven spirits of God sent out into all the earth" (Rev. 5:6).  These "seven spirits" symbolize the Holy Spirit who Christ sends in fulfillment of the longing of Moses, the promise of the Lord through Joel, and Christ's own promises (cf. John 14:26, 15:26).  

I think the Filioque is just trying to get across the point that the Holy Spirit is sent to all believers due to Christ's ascension as well as stress the essential divine unity of Father, Son, and Holy Spirit.  They are "one God in three persons."  The Son does the Father's will and together they send the Holy Spirit to witness to Christ and bring people to faith.