"Reading someone's eyes" is actually short for reading their entire face, not just their eye movements. NTs are wired to do this very efficiently and use it for a great deal of their communication. So when an NT "looks into another person's eyes" (i.e., looks at and analyzes that person's face), they can get a lot of information from that person.
On the other hand, autistic people are better off communicating in words, much of the time--there are exceptions; there are some autistics around here who say they can dump the words and use face-reading just fine, especially with other autistics--but in general, words are best. If you are feeling something, say it in words. Ask what the other person is feeling--for example, you may simply ask, "What are you thinking about?" (Just asking "how are you" will produce the form response; just use that question for protocol--it's useless for getting information about what someone's feeling.)
I think you will probably do better if you just use the date to find out as much about her as you possibly can--what she likes to do, what her life is like, what she thinks is fun or annoying. Just try to find out if you have things in common, things you can do together that are fun. Establish a connection based on facts and common interests, if you can. I would find it awfully hard to connect with a total stranger on the first date (or even an acquaintance); but then, I'm asexual and my primary need for connection is intellectual and non-physical.
Far as I can tell, that first date is mostly for figuring out whether you have potential as a couple, and whether you want to give it a shot. So just try to learn from her who she is. Get a lot of information. And if you want to invite her to a second date, try something fun that you know she likes and that you also like. Dates don't have to be all about staring into each other's eyes. Having fun together is just as good, and takes the pressure off your weaker social skills. Or maybe you will find out that you have not found a potential romantic interest, but you have found a friend you like to spend time with. That's good too.