Practice makes perfect. Get better at the game, and you will climb.
To improve, there are LOADS of things to improve upon. Last hitting is one of the easier things to practice, and one of the core things required to climb.
Last hitting can be practiced in custom games. First just pick a champ and go to a lane and try to not miss a single CS.
http://leagueoflegends.wikia.com/wiki/Farming
At the bottom there is a chart for max CS.
Once you've started to get the hang of that, gradually increase difficulty of doing this. Add bots for distraction, pick a harder to farm with champion, do not use runes/masteries.
You can also focus on many different things like freezing, pushing etc.
IF you really have patience and REALLY want to improve I reccomend this youtube channel.
https://www.youtube.com/user/unswlolsoc
They explain pretty much ALL concepts of league IN DETAIL. It's university level lectures about the game. REALLY worthwhile to watch, REALLY educational, but they are very long as well.