Christianity addresses the problem of evil by affirming that while God is not the author of sin, He sovereignly ordains all events for His holy ends, bringing good out of evil to display His glory and justice. While Reformed traditions vary on the mechanics of providence, all affirm that God is entirely righteous even when human and demonic agency act unjustly.
The Reformed tradition, as expressed in the 1689 LBCF Ch.5 §4, maintains that God's providence extends to all sinful actions of men and angels without making Him the author of sin. This means that while evil acts proceed from the creature's own wicked will, God bounds and governs these acts to accomplish His own wise and holy ends. This is not a 'bare permission' but a manifold dispensation of divine government. As Institutes Book 1 Ch.18 §4 clarifies, there is a vast difference between God's secret will and His revealed precept; while God does not command evil, He justly uses the evil intents of wicked men to execute His righteous judgments.
The ultimate Christian response to evil is not found in abstract philosophical explanation, but in the person and work of Jesus Christ. As stated in The City of God Ch.9, God's infinite goodness is such that He allows evil to exist in order to bring forth a greater good. The cross is the historical pinnacle of this mystery: the greatest evil—the murder of the Son of God—was used by God to secure the greatest good—the redemption of sinners. We rest in the promise that "to them that love God all things work together for good" (Romans 8:28). You are more sinful than you dared believe, but in Christ, you are more loved than you dared hope, for even your deepest suffering is under the sovereign care of a Father who turned the darkness of Calvary into the light of resurrection.