I'll start off with a compliment. The first paragraph was very well-written. You've got some interesting lines, all of which are incredibly poetic. It's a good way to set up the dramatic, romantic tone of the rest of the piece, and it gave the reader a good glimpse of who Elensa is by letting us hear her speak.
From there, though, it could use a bit of polish.
To start off, a lot of this fic is centered around telling instead of showing. Rather than show us interactions between Narum and Elensa, you tell us Elensa got a message, Narum's father disapproved of their relationship, that sort of thing. We don't get a glimpse of Narum breaking the news to Elensa that his father forbade them from meeting. We don't see Elensa crying or watch her talk things over with Narum. It's all basically Elensa telling us these things way after the fact, rather than taking us through each scene, at the risk of being redundant about it. As a result, it's harder to connect emotionally with Elensa because we don't really watch her struggle. We're only told that she does.
Also, telling instead of showing their plight has another negative side effect. Most of the fic seems to be focused on sounding dramatic, which makes it a little harder to take it seriously. This is mostly because it seems a little on the narmy (i.e., unintentionally funny) side when you start off with a very serious and dramatic paragraph beginning with a pretty cool metaphor and then reveal that their relationship takes place primarily over a game. Then, we find out that the players are just starting high school (by the end of the fic, at least), yet they're talking about each other like they're soulmates. What finishes this off is the entire trouble involving Narum's indecision. ("I might not love you. No wait! Yes, I do!") That last issue seems to have lasted only about five minutes because there's really nothing more to that discussion besides Narum's two lines. Sure, you tell us that Elensa goes through some angst and that Kimi helped the both of them through it, but we don't really see the argument happening. We don't really feel the sting of tension. We don't watch Elensa get more and more frantic and Narum get put off by the fighting. It's just "I might not love you" and "I need you."
As a result, the relationship ends up feeling a little shallow. From the looks of it, it's an internet relationship between two kids who barely know what love is like beyond typical romance novels and flicks. Their only trouble is Narum's father, who's justified in being cautious and preventing Narum from just running off to be with Elensa. (I mean, Narum is pretty young, there's all sorts of shady people online, and it's very easy at that age to mistake a simple crush with the full-fledged, "I need you" kind of love.) Moreover, it's difficult to take all of this seriously because, put extremely bluntly, you emphasize the drama of love so much that there's not a paragraph that goes by without Elensa telling the reader that love is difficult or that people in love can overcome all kinds of odds.
To put it shortly, what you need to do is the same kind of thing you'd do if you used third-person POV. Even if a character is telling this story, you'll still want to let us see them interact with other people. You'll want to show us scenes, their conversations, their actions, and how they feel at that exact time. That way, the relationship seems more in tune with reality. It wouldn't seem like it's happening in five minutes, and we'd be able to pick up on the emotions of the characters.
Also, don't try to emphasize any bit of drama too much. The more you say "love will conquer all" or "we were two people against an uncaring world," the more it feels like a parody than it does something you meant to have be a serious romance.
Finally on the concrit front, a couple of nitpicks:
1. Always end a sentence with an ending mark of punctuation. For example, you're missing a period after "nothing to hide" and "afraid it would be too much."
2. This may simply be personal taste, but don't insert comparisons to Romeo and Juliet in your romantic work. It's cliché, and in any case, it's not a good comparison. Romeo was literally a stalker, and Juliet was an idiot who was both below the age of consent and the entire reason why their relationship ended in a double-suicide. The both of them barely met when they fell in lust (not love) with each other and decided to get married despite not knowing a thing about their soon-to-be-spouse besides what their last names were. To finish things off, yeah, double-suicide. Shakespeare's tragedies tend to not be so much about love but more about stupidity induced by love. (And on that note, Othello is a better tragedy, complete with a far better example of a relationship. Even though that ended badly too.)
Long rants short, I can tell you have talent. You can set up a nice mood, and you can find fitting words for a scenario. The only thing you need to do is remember that you want to focus more on storytelling and character development in any story than just the emotion you want to evoke.