On Writing Sympathetic Morally-Ambiguous Characters

First, your morally-ambiguous characters need not, and probably should not continuously angst and bellyache over each and every less-than-spotless action. For example, things such as stealing an item of power from someone who would mainly have used it to cause harm to others even if it did rightly belong to that person or breaking a few ribs on a security guard whose only crime is doing xir job might not bother the character's conscience too much. A "gray" hero who felt guilty and remorseful over every questionable action taken would in reality suffer a mental breakdown due to the cognitive dissonance at some point, or quickly learn to start rationalizing and justifying the actions so that they no longer weighed in on xir conscience.

However, the character should have to grapple over major decisions where the implications of each choice are understood by the character. For example, is it worth sacrificing the lives of several innocent people to save the life of one man on the verge of a medical breakthrough that could save more people… but only because he exploited other researchers and stole their work? Which would be the worse evil: forcing your only daughter into a loveless marriage, or letting the people in your kingdom face starvation or slaughter because you couldn't otherwise secure the alliance you needed to gain assistance to avert these crises?

Even if there are no "right" choices in these scenarios, they should still present difficulty, and even haunt the character into the future depending upon how things go. If your character never has doubts or second thoughts over anything, xe will look utterly self-centered and heartless, which does not make for a sympathetic character.

Do not try to force the audience's perception of your character, nor coerce the audience into taking your character's side with justifications for your character's actions while showing that those who opposed your character's actions are clearly in the wrong. If the audience doesn't feel too positively about a character's actions already, trying to force them to like the character again will mainly make them hate the character more.

A character's moral ambiguity should never be treated as a license to do whatever xe wants without consequences or censure, nor should you start thinking of it as a reason your character is better or superior to others. Let other characters get genuinely angry with them upon occasion, especially the ones who end up shafted by your character. Let it be shown that these characters actually have legitimate reasons to be angry, and that they aren't simply being self-centered whiners because they aren't willingly and ungrudgingly giving of themselves and theirs. Let there be repercussions to your character's actions. (And remember, protagonists who are so morally myopic that they only care about themselves aren't "morally gray" - at best they're just plain selfish assholes; at worst, they're outright villains.)

If your character comes out of each and every situation smelling like a rose as far as anyone who matters is concerned, you don't really have a morally-ambiguous character. Instead, you either have a character who never really had to make a sufficiently dramatic decision to be considered morally-ambiguous in the first place, or you have a character whose actions are above reproach for the simple fact that it's this character doing them. Either way, you're doing it wrong.

Long story short, a sympathetic morally-ambiguous character needs to break rules and make decisions that others might never agree with, but at the same time retain xir humanity. It must be shown in-universe that the character's actions will not be well-received by all (including people whose opinions matter), and if appropriate, there should be negative consequences for the character.