Characters Must Make Sense
If you want to write believable, compelling characters, their motivations must be understandable. They must be doing things for reasons the reader would do things, or at least for reasons that make sense to the reader.
"Some guy killed my puppy, so I'll kill him," makes sense, even if it may seem disproportionate. "Some guy helped me out when I asked, so I killed his puppy," doesn't make much sense; it's just arbitrary.
Moral Ambiguity Usually Arises From Complex Situations
Murderers aren't good people. This is a fairly unambiguous statement. And yet, there are (usually not very good) stories about assassins who are heroic and sympathetic. A tragic backstory about an abusive childhood, taking care of a needy sibling, running out of alternatives, being forced to take on an opportunity to do something terrible for profit... (And maybe making the victim seem deserving of death.) All of this can contribute to making an assassin seem more relatable.
We know, from experience, that people sometimes do bad things. We may have even, in a moment of anger, wanted to do terrible things ourselves. But to get the reader to believe a morally questionable action was reasonable, or at least sympathetic, generally requires a situation where there is no better alternative. It will not be compelling if a character is bored of a wholesome and prosperous life, and starts killing the friendly and supportive people around him just to spice things up.
What In the [Real] World is Ambiguous?
Saying that you want to write a story with a morally ambiguous character is like saying you want to write a story with dramatic action scenes. Are we talking about exciting magical battles, gritty historical wars, or a mugging?
What actual moral ambiguities do you want to write about? What actual (fictional) situation is making the character's decisions seem reasonable from one perspective, but unreasonable from a different perspective?
An assassin working to support his poor siblings so they can have a better (less tainted) life is sympathetic from the point of view of caring for your family. But is he causing more harm, more grief, from a broader perspective? Do the new orphans made by his killing have a different perspective? Is loyalty more important than kindness or mercy? And mercy to whom?
Moral ambiguity requires an inherent contradiction in values, and preferably a believable contradiction. To persuade your reader to doubt what is right, you must have a specific contradiction (or more than one), which shows that "right answer" isn't always right.
Or you could just have a really charismatic character who sometimes does bad things, but you really want to like her anyways... I wouldn't recommend it, though.