Paul begins his discussion in Romans 5 with the following fine affirmation: Therefore, since we are justified by faith, we have peace with God through our Lord Jesus Christ, through whom we have obtained access to this grace in which we stand (5.1-2). This new peace with God, in light of the previous enmity and darkness of sin apart from the grace of God (cf. Rom 3.9ff.), offers an entirely new vantage point from which to live life.
For instance, Paul goes on to say that Christians rejoice and boast even in their sufferings (v. 3). How on earth could this be? In the first place, it seems empirically false -- after all, how many Christians do we know that do not rejoice but rather complain in their sufferings, even eventually abandon the faith altogether? Yet Paul seems to affirm rather confidently that suffering produces endurance, and endurance produces character, and character produces hope (vv. 3-4).
I think it would be mistaken to understand Paul's language here as describing some kind of automatic process. The point is not that if you're a Christian, suffering and hardship will automatically produce hope, character, and endurance in you. We know that this is generally not true; the natural reaction to hardship, at least for many people, is despair and a kind of spiritual deflation. Presupposed is that the Christian voluntarily and freely engages in hopeful endurance, that the Christian makes use of suffering as an occasion for developing character.
My proposal is that Paul understands the Christian to do all this because of her new standing before God, thanks to Jesus Christ. Precisely because we have been justified by faith and now have peace with God, we can and ought to approach suffering and hardship differently than before. We find in hardship an occasion for soul-making, an opportunity to develop our characters differently than we had previously done.
And we don't have any trouble hoping in God, because God's love has been poured into our hearts through the Holy Spirit that has been given to us (v. 5). We love God, we know that he has done something fantastic and great for us in Jesus Christ, and so we can trust him. Paul later poses the question: What then are we to say about these things? If God is for us, who is against us? (8.31) Indeed, if God is for us, then what can mere sufferings and earthly travails do to us? On the contrary, by the power of God we can turn them into occasions for making ourselves more like Jesus Christ.