Personally, I'd think it'd be for the best if you don't. Both because her character wildly shifts from author to author (much like the rest of the gods) and because it's bad for you health.
Now, that being said...if you choose to write the Greek Gods as more grounded, then it's completely workable.
First, make sure Zeus likes you. It's honestly easier than you'd think if you pop in at the right time, and as long as you continue to offer sacrifices on the right days, respect Xenia, and generally be a decent person you're not likely to loose that good opinion of you. Also, it'll just be helpful in general.
Then, befriend both Apollo and Artemis. Great hunting skills and being a good, friendly person should work for Artemis, though don't go in looking to romance her. This'll make the whole process more genuine, and help with Apollo.
As for Apollo, just be good at one of his various skill-related domains, offer sacrifices at the right time, and don't claim to be better than him. In fact, if you claim your talents arise from his blessing, he'll appreciate that a lot. Even if in this case it isn't true, it is how talent and inspiration usually work here and mortals are bad at acknowledging that consistently.
Also, be a god of some sort.
Once you and Artemis fall in love, subtly suggest she asks Zeus for the right to marry before actually doing anything. If both Zeus and Apollo actually like you already, while they won't be happy you and Artemis are in love, they're fairly unlikely to actually do anything about it if you pulled it all off.
Just make sure to actually get married quickly after Zeus gives his approval. The less time you give for Apollo to work himself up to doing something about it despite himself and his father the better. Once you're actually Artemis' husband, you don't have to worry about Apollo trying to kill you.