A person executed for murder in the 1800s would have been put to death following a certain procedure.
First, having aroused sufficient suspicion, the accused would be imprisoned in the nearest suitable jail or prison. Then, a hearing ("rannsakning") would take place in the local courts, with witnesses, evidence etc and the local kronolänsman as prosecutor. If the häradsrätt found the accused guilty, they would as a rule ask the upper jurisdictions to comment : hovrätten or even the king personally.
This means that executions leave a decent paper trail which one can follow. I recommend Justitiekanslerns fånglistor (list of prisoners), sorted by län, as a good first step. The first step in any criminal case is obviously the local court records but the digitisation coverage for 1800s Sweden is still spotty at best. Arkiv Digital are currently digitasing JK:s fånglistor and they should have full coverage for the years in question.