Is it ever appropriate for a Christian to bring a lawsuit against someone for defamation of character?
While the Bible doesn't comment specifically on the issue of bringing a lawsuit for defamation of character, it does have some other principles which I think will answer your question.
First, the Bible teaches that Christians are NEVER to bring a lawsuit against another Christian. Paul writes in 1 Corinthians 6:1-8 that to do such is sinful and fraudulent. Paul states that we should rather suffer harm than bring a lawsuit against another Christian, but if that there is a dispute that cannot be settled, that it is better for another Christian to decide the matter privately. So it would be a sin for a Christian to bring a lawsuit against another Christian.
Second, as far as bringing a lawsuit against a non-Christian, it would depend upon the circumstance. When Paul and Silas were unjustly cast into prison, Paul demanded that a public acknowledgement of the wrong that was done to them be made (Acts 16:36-38). This was due to the fact that Paul was a Roman citizen and had certain rights as a Roman citizen that were inviolable. This would be similar to our right to free speech taken away in this country. If that was done merely because we were Christians, then it would be appropriate to pursue that under the law. I don't think that this applies to your situation.
In regard to defamation of character, that would be similar to what Peter warned early Christians to do in such a situation in 1 Peter 2:19-24. Here Peter says that if others speak evil of you as a Christian and you take it patiently, then you have followed the example of the Lord Jesus, who, when he was reviled, did not revile again. Also note 1 Peter 3:16-18 as well. Both of these passages teach that when we are wrongly slandered, that we should not return evil for evil, but good. This comes nearer to addressing the situation of slander and defamation of character than anything else within the scriptures.