I think that in order to know what righteous anger might look like, we will do well to look at the example of Jesus. Clearly Jesus got angry, yet we know that he did not sin. Two examples come to mind. The first is when Jesus cleared the temple. This is recorded in John 2:13-16. Here Jesus is clearly angry that the Jews have made the temple grounds into a profit opportunity–caring nothing for the purpose of the temple, which is to give glory to God. What I see here is that Jesus is not angry at what was done to him but to the honor of God. He did not lose his temper, but rather acted slowly and deliberately. He did not use violence on people, but simply used a whip to drive the animals out of the temple area. We know that he overturned the money tables. This is a clear sign of anger. From this incident I learn that righteous anger will normally be inspired, not by something done to us but something done to God or to someone else. If you or I are angry over something done to us, it is quite a bit less likely to be righteous anger.
Another example comes to mind. Jesus was obviously quite angry at the Pharisees in the scene we find in Matthew 23:1-36. Here Jesus is clearly frustrated and angry at the hypocracy of the Pharisees. He calls them whitewashed tombs and what they were–hypocrites. Again, although we see angry words here, we do not see personal attacks. Jesus is passionate, but not out of control. As with the first scene from John 2, Jesus is angry over the fact that the Pharisees have abused someone other than himself. In this case, they have hurt their hearers by being a very bad example and by leading people away from God rather than toward him.
We find the apostle Paul angry on more than one occasion. He was very angry at the Judaizing "Christians" who tried to force Gentiles to obey Jewish laws. He says of those who preach a distorted gospel, may they be eternally condemned (Galatians 1:8) and may they go ahead and castrate themselves! (Galatians 5:12). These are very strong words. I hesitate to use Paul as an example of righteous anger, as we have no assurance that he was without sin. For all we know, Paul may have been in sin when he penned these words. More likely, this is righteous anger, given that it made it into the Bible. Again, Paul is angry, not at what they did to him (and they did plenty of bad to Paul), but at what they were doing to other people–in some cases even keeping them from going to heaven.
To summarize, it seems that righteous anger is directed toward people who willfully violate the rights and prerogatives of God or other people. Anger at those who commit acts of racial prejudice or those who scoff at God or who create division in the church is appropriate. Anger at one’s family members for taking advantage of us, probably the most common cause of anger, is not in this category. A parent’s anger at their children for blatant rebellion may be appropriate, but that depends on whether it is expressed in a righteous way. If such anger results in a desire to hurt or get revenge it is not righteous. Even in situations in which righteous anger might be appropriate, it is very common for us (and I include myself, by the way) to hold an kind of anger which is not righteous.
It is my experience that most people who ask about righteous anger do so in order to find a way to justify their own unrighteous anger or to prove to someone else that their anger is not righteous. I trust that this is not the case with you! So, if you want to know what righteous anger over the sin of another Christian looks like, the first thing I would ask is whether that sin was against you. If it is, you probably will do well to assume that your anger is not righteous. If it is anger over a sin against the honor of God or another person, you still ought to weigh the feelings and actions against the examples of Jesus.