How Do God’s Love and God’s Wrath Relate?

One evangelical cliché has it that God hates the sin but loves the sinner.
There is a small element of truth in these words: God has nothing but hate for the sin, but this cannot be said with respect to how God sees the sinner.
Nevertheless, the cliché is false on the face of it, and should be abandoned. Fourteen times in the first fifty psalms alone, the psalmists state that God hates the sinner, that His wrath is on the liar, and so forth. In the Bible, the wrath of God rests on both the sin (Rom. 1:18–23) and the sinner (1:24–32; 2:5; John 3:36).
Our problem in part is that in human experience wrath and love normally abide in mutually exclusive compartments. Love drives wrath out, or wrath drives love out. We come closest to bringing them together, perhaps, in our responses to a wayward act by one of our children, but normally we do not think that a wrathful person is loving.
But this is not the way it is with God. God’s wrath is not an implacable blind rage. However emotional it may be, it is an entirely reasonable and willed response to offenses against His holiness. At the same time, His love wells up amidst His perfections and is not generated by the loveliness of the loved. Thus there is nothing intrinsically impossible about wrath and love being directed toward the same individual or people at once. God in His perfections must be wrathful against His rebel image-bearers, for they have offended Him; God in His perfections must be loving toward His rebel image-bearers, for He is that kind of God. . . .
The reality is that the Old Testament displays the grace and love of God in experience and types, and these realities become all the clearer in the New Testament. Similarly, the Old Testament displays the righteous wrath of God in experience and types, and these realities become all the clearer in the New Testament. In other words, both God’s love and God’s wrath are ratcheted up in the move from the Old Testament to the New. These themes barrel along through redemptive history, unresolved, until they come to a resounding climax in the Cross. 
Do you wish to see God’s love? Look at the Cross.  Do you wish to see God’s wrath? Look at the Cross. —D.A. Carson, “God’s Love and God’s Wrath,” Bibliotheca Sacra 156 (1999): 388–390.