With Greek Gods, I would begin with understanding Greek Religion, and the modern practice of it. "Hellenismos" and "Hellenic Polytheism" are some good key words to search for in Google. I haven't really seen too much of a historical practice of personal patronage when it comes to Greek Gods. Usually, they watch over groups,places, or activities, not idividual persons.
The reason I'm saying this is because most people in the West know about basic Greek Mythology, but we are not taught about the religion from the culture. This would be part of the reading as H20 suggested.
there's a bunch of ways. if you're interested in Greek mythology, research the deities and if one stands out to you, try connecting with them. [you could also narrow it down by looking up deities who rule over something you love like owls, winter, or agriculture. you could also look into deities associated with a specific job you do say medicine, building, or even home and family] once you found the deity you wish to work with, try setting up an altar in their name and leaving offerings for them. meditate by it daily, and pray to them. you should receive an answer within the month.
sometimes the deity won't respond or will only stay for a while, accept this and keep on your path. you will find your deity in time. sometimes the deity will reach out to you, so keep an open mind. i found Ember HoneyRaven really helped me on my path and i recommend checking out her video.
Just to point this out, as I believe you were doing I just want to push it further, you do not need to be "called" by a deity. You do not need to receive an answer. You don't have to have this super personal connection to a deity in order for you to give them worship. A lot of the Hellenics I interact with love to say along the lines "the answer is yes, and the Gods appreciate your worship."
Just a reminder that you do not need to be a theologian or a spiritual worker to follow and practice any religion. You can just be.
The honest answer being you do research. What a god is portrayed as in pop culture is often about as accurate as how they portray magick. Many times I have seen pagans as well as gods portrayed as villains. So many inaccuracies be it in Wonder Woman's story or in The Avengers Thor. So something you once might have been put off about you might be very surprised to find was nonsense added to get ratings.
Google is your best friend when it comes to looking into things. A lot of the stories whether you believe them true or not I feel still speak of their character. So a bonding experience if you will. It is also a good way to find out what said god likes or dislikes. Such as how Athena might appreciate crafts, Aphrodite a pink rose or yellow apple, etc.
Here is a link that might help you in your journey:
http://www.theoi.com/greek-mythology/olympian-gods.html
