John Chrysostom, one of the Early Church Fathers, wrote, in his Homily 10 on Second Timothy, that Paul was referring to his trial when he said Alexander had done him much harm. The next verse relates that he (Alexander) greatly withstood Paul's words, opposing his message, and the verse after that is clearly about Paul's trial , saying he had nobody to support him during it. On this view Alexander was a principal and effective opponent of Paul and probably a witness, and major complainant, against him. He may also have turned others against Paul, perhaps being the cause of the lack of support for him which he mentions in verse 16. Gill's Exposition suggests Alexander had done great injury to Paul's character and had reproached and reviled him as a man of bad principles and practices.
It is likely Alexander was from Ephesus, and may have travelled to Rome expressly to cause difficulties for Paul. Timothy is thought to have been either at Ephesus, or on his way there, when Paul wrote to him which suggests he (Alexander) may have returned there.
Many commentators suspect that this Alexander the Coppersmith was the same Alexander mentioned in Acts 19 v33. This was at a riotous assembly in Ephesus provoked by Demetrius, a silversmith. Demetrius was concerned that Paul's preaching was damaging their trade, causing a slump in demand for religious artefacts associated with the temple of Artemis. Demetrius gathered together other metalworkers to protest Paul's damaging influence on their trade. The word "Coppersmith" doesn't just mean coppersmith, it refers to any manufacturer of metalic products, and so Alexander the Coppersmith would likely have been involved. A large and tumultuous gathering resulted, many participants not knowing what was going on or why they were there(verse 32). This Alexander was then thrust forward by the Jews and attempted to speak to the crowd, gesturing them to quiten down so that he could make a speech. He was presumably seen as a potentially powerful orator, which again suggests he may be the same man as our Alexander 'Smith. We do not know what he wanted to say, partly because he never got to say it. Instead the crowd began chanting "Great is Artemis, Great is Artemis", and continued doing so for two hours until eventually the Town Clerk succeeded in dispersing them (verse 41).
The fact Alexander didn't, in the end, get to make his speech has not stopped later generations from speculating about it. Calvin, in his commentary, suggests that Alexander was a Christian believer and intended to speak in defence of Paul and of Christianity, and that he was willingly risking martyrdom by attempting to do so. Chrysostom, however, wrote that Alexander was attempting to inflame the matter. Matthew Henry suggests that the Jews who thrust him forward wanted Alexander to denounce Paul as an enemy of Judaism, as well as of Artemis and the metal industry.
If Alexander was not a Christian his animosity for Paul may have begun there.
Another mention of the name Alexander occurs in 1 Timothy chapter 1 verse 20. This Alexander is a pparently a renegade Christian believer who Paul has "delivered to Satan" for blasphemy, speaking against the truth. If this is Alexander the Coppersmith then the animosity between the two men may date from there, if not before.
Paul's remark that God will recompense Alexander Coppersmith may simply mean Timothy should not attempt to do so himself. Leave it to God. Paul is warning Timothy to be careful of him.