I'm going to ignore all that **** about your item builds right now that everyone else is talking about and look at something far more basic, and that's your basic ability to last hit. Looking at your match history, your last hitting is abysmal. In your last ten matches, you only achieved higher than 100 cs (creep score, the number of minions you killed) in 4 of your matches. In only 2 of those did you get more than 150 cs. In 3 games, you got less than 50. That's terrible unless you're playing someone like Janna, who tries to give their lanemate all the cs as a support (but... that's not Kennen, Akali, or Anivia). Focus on how to last hit better - wait for minions to get low on health, then deliver the killing blow. Farming well is essential to gaining a lot of gold. It doesn't matter if you finally build your core of Rylai's and Lichbane or whatever if it takes you 40 minutes into the game to do it. Every 15-20 minions you kill is basically like adding another +1 to your kill count. By last hitting so poorly, you are essentially throwing away a lot of potential gold. A poor assassin with no items is, well, a poor assassin.
Getting kills early in the game is secondary to farming a lot of gold via minion kills and if you don't farm well, naturally your damage will be terrible because you have no items to back yourself up with. In a solo lane, a decent last hitter should relatively easily get around 100 cs at around the 11-12 minute mark (you can actually get 100 cs by the 10:00 minute mark if you're perfect). At this point you have gotten an average of 23 gold per melee and 17 per caster creep (gold per creep actually increases slowly over time). Assuming you got 50 of each, that's 1150 + 850 = 2000 gold potential extra gold at the 11 minute mark, and with the natural gold over time, that equates out to around 2,800 gold. In some of your games, people have earned almost as much gold in 11 minutes as you have gained over the course of the entire game. That means they hit a lot harder a lot earlier, when defenses and health are still low. People are walking around at the 11 minute mark dealing out as much damage as your are at the 25 minute mark, and by the time the 25 minute mark comes around, you can ensure that they're far more powerful than you are. If you want to be better, last hitting is an essential skill. In a 1v1 between a person that is 3/0/0 with 50 CS and a person that is 0/0/0 with 150 cs, I'd probably put my bets on the 0/0/0 person if the champion matchup is reasonable.
Farm your items first, wreck faces later.
Now looking at your items, you seem quite confused as to what to get. All your recent games are played as Akali, so I have no idea what you do as Kennen or Anivia, but in 3 of your games, you didn't even activate the AD portion of Akali's passive. In a couple, I imagine you didn't activate it until mid-end game because your only source of AD comes from Pickaxes (Gunblade or Guinsoo's Rageblade), and even the games with Trinity forces are kind of questionable as with the positions of the items I kind of doubt you started with a Longsword. That's pretty bad. The spell vamp given by the AD portion of Akali's passive is extremely helpful for staying in lane a bit longer and absorbing some harass.
For chasing, Rylai's Crystal Scepter is a very common choice. It gives her an amazing ability to stick to a target (Q and R proc the stronger single target slow), and it gives her a decent health buffer as well. If you fight in your shroud, you become relatively tanky mid game.
If you have a lot of AP, go Lichbane. It's better than Trinity Force for her, especially if you're building more AP via items like Rabadon's Deathcap or even Hextech Gunblade. The 1:1 AP ratio and addition AP from the Blasting Wand will be more useful than the critical chance and 1.5:1 base damage ratio (base damage means that it doesn't take into account damage from items like a B.F. Sword), and if you get a Rylai's, the slow from Phage is completely wasted because item slows do not stack.