I want to create a log of my quests completed as well as describe some of the things that happened during them and sort of describe them in story format. Not in the typical kill 12 things and get X but rather cull there numbers and when a noticeable change occurs you will have proven yourself worthy of X. Also it would be sort of from a battle to battle type of thing instead of "after i killed 12 of x then...." What I want to know is if A) That would be allowed and B) If it is allowed, exactly where would I place it. I'm guessing in storylines but I'm not exactly sure.
Quests are generally not considered 'IC', because they are fairly unrealistic. Even the lowliest creatures, say... Murlocs, could overpower somebody eventually. The quests would have you believe that a single warrior can easily waltz into a village of ...., murder 20 of their members, kill their leader and collect some paws, but that's just plain unlikely. Even the bigger lore figures aren't exactly invincible juggernauts who can pull that off, which makes using the quests for your character's history a bit odd.
A Perfect example is that quests would have you believe that you can kill Illidan or the other raid/Dungeon bosses.
I think what you are describing is in fact a bit like journals written by your character describing events in their past, through RP, or just daily life. While not exactly the quest idea you stated, it is generally the same idea and to a degree is encouraged to the development of your character. Obviously the above two posters hit the right notes, but in regards to simple tasks that a man could reasonably perform there is no limitation to what you can create, and a good writer can turn a simple thing into something really interesting.
Ahem... anyhow, if you want to create a journal describing the events of your character ICly, then it would fit in the Storyline section, and I think that'd be a great idea. To get a feel for what I mean check up on some of the threads in there and you will find a wealth of inspiration and some rough guidelines on what makes a good IC post and what doesn't.