I'm not aware of any official ruling on the topic, so you (or your DM) would have to fall back on common-sense reading.
If I point the bullseye lantern away from me, is my character illuminated in any way? Am I included in the cone of light, or am I outside of the cone, like when casting the Cone of Cold spell?
I think clearly you're outside the cone of light. That's the how the lamp works, assuming a perfectly assembled lamp that doesn't leak light in any direction other than out the shutter.
If I remain in darkness when using a Bullseye Lantern, then I can remain invisible to those without dark vision, while illuminating my enemies.
Darkness doesn't make you invisible in the first place. So it's not clear what you mean by "remain invisible". You can't "remain" something you aren't to start with.
If you are actually invisible, then whether using the lamp would terminate your invisibility or not depends on the source of the invisibility (which, as established, we already know isn't the darkness…might be a spell, or a cloak, or whatever). But generally, invisibility is, at worst, terminated by an attack, and using the lantern wouldn't normally be considered an attack, so if you had actual invisibility, the lantern shouldn't affect that, even as it would reveal your position.
Note that I'm assuming here that you're playing with the interpretation of the interaction of light sources and invisibility where light sources held by an invisible creature still emit light visible to others. There is no clear consensus on this, especially since Jeremy Crawford's tweets no longer are authoritative for rules interpretation. You could assume that invisibility prevents the lantern from being seen, but then it would also be of no use and irrelevant to the question you asked.
Now, all of the above said…
First, note that the description of the lantern doesn't even tell you the exact dimensions of the cone, except its length. Spot lights are generally modeled in computing as a zone of full brightness, shaped like a cone, with a larger cone around that where the light falls off, and then finally with complete darkness anywhere outside the larger cone.
Even using that idealized version of a lamp, you have to decide what the angles of those cones are, something not provided to us. So, make some assumptions. Say, the inner cone is 45 degrees and the outer is 90.
More relevant though is your environment. Take a very bright flashlight outside in the dark, and shine it straight up in the air. You won't be illuminated at all by the flashlight. On the other hand, if you shine it on the ground, then depending on what's actually on the ground you will be illuminated to some degree. Lighter colored surfaces will illuminate you more than dark surfaces, but in most cases there will always be some light that bounces back onto you.
So, are you in a small limestone cave when you do this? Then it is likely the light cast by the lantern will be reflected by many surfaces around you, illuminating you fairly well.
On the other hand, are you outside on a plain of black lava? If so, then little or no light is going to be redirected toward you, and you'll remain unilluminated. Though of course, if your opponent has a shiny shield and positions it to reflect the lantern's light back on you, you might wind up illuminated after all.
So the exact effect of the lantern is situationally dependent. If something happens where it matters how well anyone can see you, your DM will have to determine based on what's actually going on and where you actually are, how that all plays out.
Noting, of course, that merely using the lantern gives any other creature a clear indication of your position. Even if your body is coated in 100% light-absorbing black pigment, unless you are dangling the lantern from a pole well away from your body or something, it's not going to be hard to infer where an attack ought to be directed for a reasonable shot at hitting you.
If you are in fact completely light-absorbing, then I'd rule that you blend in with the complete darkness around you, rendering you effectively invisible, giving attacks against you disadvantage (and your own attacks advantage). The lantern in that case acts similarly to your footprints in sand exposing your location.
But otherwise, the more likely outcome would be something akin to applying the rules for cover, with a +2 or +5 bonus to AC, depending on just how much light has bounced back on you.