How do you feel about your own sense of empathy?
I'm sure I have a healthy sense of empathy.
Do you feel lacking or do you feel it is one of your strengths?
I feel that it's a confusing subject. It can be a strength but it's heavily context dependent.
Do you think the common perception is accurate or misleading?
I don't merely think it's misleading, I know for a fact that it is. Popular culture is pitifully far behind the zeitgeist, as usual. Popularly speaking the word "empathy" is conflated with compassion, and the distinction between cognitive empathy and compassion is blundered over.
A few things to keep in mind:
-A person's theory of mind doesn't have to be average or above average for them to be compassionate, and/or have sympathetic or somatic responses to perceived emotions and circumstances of others.
-Having ASD may leave us at a disadvantage in developing an adequate theory of mind, i.e. comprehension of the probably emotions experienced by people around us in any given situation, however what passes for "adequate" is hardly what it's vaunted to be. People tend to be very confident about their perceptions but experience tells us, if we're willing to observe, that everyone is wrong all the time about the emotions that others experience.
Theory of mind serves as a social roadmap (at least this is one of, if not the most, important component of it's usefulness), helpful, but hardly scientific and I may be a broken record here in saying this, but it can really never be restated enough that an infinite void separates every two people. My red may very well be your green for all we know. My happy, sad and angry are more than likely dissimilar to your analogous states of mind.
-Expressing empathy and having it are also two distinct things. Judge couldn't have stated it better in the first response to the OP.
IMO, the most critical part of this issue is that it is not that people on the spectrum necessarily lack empathy, so much as that it can be a struggle for us to project it in a meaningful way that Neurotypicals can understand.
Just because we don't communicate it in a conventional way, doesn't mean that we don't have it. Perhaps one of the most critical misconceptions that exist between the Neurotypical and Neurodiverse.
While it's easy for me to express myself in text here, projecting empathy just isn't as easily expressed verbally in person in real-time. For whatever reason the words aren't always there...but the feelings are.