I started with the characterization developing in my mind -- imagined conversations between my character and a follower during the walks/runs across the face of Skyrim --, and from there I found moments that leapt out as pure narrative, narrative existing beyond that of the scripted Skyrim events and quests. In such moments, I'd crack open a Word file (or actually, in my case, an OpenOffice document) and type up five or so hundred words -- a flash fiction snapshot of the emotion or event that inspired me while playing.
While that is a technique I still use, I'm also a writer -- that is to say, professionally, and at my core, I am a genre fiction writer. I've spent my whole life writing and perfecting my craft, and one of the skills to which I seem predisposed, is the retention of many complex story threads and details in the absence of detailed notes. I do still take many detailed notes, and it may in fact be that a life time of taking such care with note-taking that my mind developed a greater affinity for recall, but a lot of my character's story exists purely in my mind. I've occasionally wondered how it will go when I finally (any day now) sit down and start writing the story -- will I remember emotions or dialogue? Will I remember a sequence of events? --, but I don't feel it makes much of an imapct on the narrative in the end; as long as you have a clear and consistent character concept (alliteration ftw) in mind, then whatever you don't remember you can re-fabricate as needed. Ultimately, you're the one who came up with the character and the personal events that may or may not coincide with those depicted in the game, so it's you who decides what is and isn't right for your character, and what is and isn't included in the story you write.
All this is to say that when the mood to write takes you, allow it. When the mood to play takes you, allow it. Don't worry too much about balancing one against the other. You don't strictly need to play Skyrim to write a good fan-fic for it, but as the character is essentially your fabrication, you also may not need to write out every idea, conversation, event in their personaly history, or observation they have while you play. As long as you keep the personality intact when you do write, the events will still feel natural and maintain continuity.
So don't stress, friend. ^__^ Write as you will, play as you will, it all works out.