I think you need to narrow down you champion pool a bit. If your sole goal is climbing then you should only be playing 2-3 champions per role and focus on mastering those champions.
Same can be said for roles. In your past 20 ranked games you've played every role except ADC. Its certainly possible to climb being a jack of all trades but its going to take you a lot longer.
CS/min looks about average but definitely could be a little higher. Pink wards, damage to champions, etc all look good.
KDA seems to be your biggest issue. If you look at every single champion you play with >3 KDA you also have a positive win ratio. Once you drop below 3 KDA your win ratio becomes more shaky. I've noticed the exact same thing in my games. As long as I average >3 KDA per game I tend to have a positive win rate and I'll climb. What this translates to is that whenever you die you should think "how could I have prevented that death?" Over time you will stop making those mistakes that are causing you to die. If you are able to die 1 or 2 times less per game I bet you will see your win rate improve enough that you'll climb.