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Source: https://writers.stackexchange.com/a/38251
License name: CC BY-SA 3.0
License URL: https://creativecommons.org/licenses/by-sa/3.0/

These are the factors that influence where the line is. It will be floating, depending on context. The following would increase reader sympathy, given a morally dubious main character:

1. Motivation - Give the character a compelling reason for their reprehensible actions.

2. Information - Create a scenario where the bad actions are the result of misinformation the character receives.

3. Comparison - make other worse characters to compare your character against. You can even make your main character antagonistic to them.

4. Feedback - produce negative repercussions for your main character's bad behavior.

5. Enjoyment - as other answers have noted, a character that enjoys doing bad things has severely damaged relatability.

6. Flaws - a subset of (1), your character may have tragic or otherwise blameless reasons for being morally flawed.

7. Context - maybe your main character's bad actions are, relative to other options, far superior. The executioner who kills torture victims may personally loathe their work, but consider himself to be a merciful figure. 

8. The victim - make the victims even worse people, a subset of (3). We root for Dexter not because he's a murderer, but because his targets are (arguably) acceptable targets of violence.

Original score: 5