In the City of God St Augustine provides commentaries on several psalms, explaining that they are prophesies relating to Christ and the Church. Here is a key extract summarising his position:
Now David was a man skilled in songs, who dearly loved musical harmony, not with a vulgar delight, but with a believing disposition, and by it served his God, who is the true God, by the mystical representation of a great thing. For the rational and well-ordered concord of diverse sounds in harmonious variety suggests the compact unity of the well-ordered city...
... Let him then who will, or can, read these volumes, and he will find out how many and great things David, at once king and prophet, has prophesied concerning Christ and His Church, to wit, concerning the King and the city which He has built.In his commentaries on individual psalms (the Enarrations), he makes this connection even more explicit. On the first line of Psalm 1, Blessed is the man that has not gone away in the counsel of the ungodly, for example, he says:
This is to be understood of our Lord Jesus Christ, the Lord Man.
Blessed is the man that has not gone away in the counsel of the ungodly, as the man of earth did, who consented to his wife deceived by the serpent, to the transgressing the commandment of God.
Nor stood in the way of sinners. For He came indeed in the way of sinners, by being born as sinners are; but He stood not therein, for that the enticements of the world held Him not.
And has not sat in the seat of pestilence. He willed not an earthly kingdom, with pride, which is well taken for the seat of pestilence; for that there is hardly any one who is free from the love of rule, and craves not human glory. For a pestilence is disease widely spread, and involving all or nearly all. Yet the seat of pestilence may be more appropriately understood of hurtful doctrine; whose word spreads as a canker...