Miss Fortune Build
Not Updated For Current Season
OK, hear me out. I realize that an AP Miss Fortune Build seems crazy, but I've had a lot of success with this and here's why: All of her abilities except for Double Up scale with AP; her ultimate actually scales with either damage or AP.
In the case of Make it Rain, the AP factor can make that deal double damage or more by the end of the game, which is extremely punishing when used as a fight starter and then followed with other abilities.
Early Game(1-6): You should be mid(if not, it's still pretty much the same although the item order is probably different). Focus more on last hitting and creep killing via Make it Rain unless your laning against someone you can kill. Start with a Doran's Ring and a Health pot. Once you get your ultimate stay in the middle as long as it's safe or there's a kill involved(you or them). Your Ignite can win a lot of 1v1 fights here if used right before you drop your ultimate.
Mid-game(~7-13): Still mid, but look for ganking opportunities as they arise. Your AoE and ultimate are really punishing against lower level champs at this point if they've overextended. You should get your boots and then immediately start working on stacking tons of AP during this part of the game. I've found that this is really the prime area for this build, so try and get as many kills and tower breaks as you can here.
End-game(14+): You've gotten everyone's attention with your AoE's and ultimates, so now the other team probably wants you dead first. Try and coordinate with your teammates so that you can be at the back of fights, away from snares and grabs. Once a team fight starts, drop an AoE and an Ignite and then lower the boom with your ultimate. Once that's over, switch to Impure Shots and Double Up, then AoE again when it pops up.
I'm still tinkering with items and the orders for them, and honestly, it depends who you're laning against in the beginning and the team comp on both sides. If you think you can get kills early, go for the Mejai's right after the boots.