My personal "skill with" list

I would recommend for anyone who's trying to get better at league of legends to familiarize themselves with the following:

Rengar (top, sometimes jungle)
Lee Sin (top / jungle, sometimes mid)
Olaf (top, sometimes jungle)
Orianna (mid)
Zyra (support, sometimes mid)
Ezreal (ADC)
Lulu (support)
Cho'Gath (Jungle, sometimes top)
Nunu (Support, sometimes jungle)
Sona (Support)
Vayne (ADC)
Corki (ADC)
Jax (Top)
Shen (Top, sometimes jungle)

That list above represents the most commonly played / banned champions at IPL (except for Diana who got nerfed hard). If you're not playing as one of them, that's fine, but you need to know how they work, especially if they're in a lane that you're typically in. Once you have at least a passing familiarity with them (at least 3-4 games played to get a feel for them, I'd recommend), figure out what YOU'RE good at. Here's the general roles in the present meta:

Top Lane Bruiser
Mid Lane Assassin
Mid Lane AoE Caster
ADC
Ganking Jungler
Farming Jungler
Sustain Support
Aggressive Support

A brief explanation as some of the differentiations are nuanced

The current meta supports two different types of mid laners. Assassins and more traditional damage casters. The latter generally has good AoE capacity (think Orianna, Anivia etc) who play very differently from the former (think LeBlanc, Diana, etc). Your goal when laning with each of them and your skill set around them can be very, very different.

Generally there are two different types of junglers at present. By far the more common is the ganking jungler (think Nocturne) but the jungle changes have given rise to another type of jungler who focuses on farming and sustaining so that they can counter jungle heavily and take advantage of their usual health advantage (fiddlesticks and warwick, for example).

Finally, we're seeing two very different support roles when you compare someone like Soraka (probably the most sustain oriented support in the game) to someone like Nunu (who's all about attacking and pressuring). You need to know the ADC you're with and what they're capable of. Unfortunately, all of the typical ADCs CAN harass, this one is more about the player. So ask your ADC what they prefer and, if in doubt, take someone like Sona that can do a bit of both.

So, for me, here's my current "competent" repertoire (note, these are my estimates of my skill level... not necessarily how good I think the champion is)

Top Lane Bruiser: Malphite > Vladimir > Nasus > Yorick > Jayce
Mid Lane Assassin: Diana > LeBlanc > Ahri > Katarina > Talon
Mid Lane AoE Caster: Heimerdinger > Veigar > Lux > Twisted Fate > Swain
ADC: Twitch > Graves > Tristana
Ganking Jungler: None
Farming Jungler: Fiddlesticks > Cho'Gath > Skarner
Sustain Support: Soraka > Sona
Aggressive Support: Sona > Blitzcrank > Nunu > Zilean

Those represent my comfort level with a variety of champions. Note: The lists need to be read independently as I'd feel much more comfortable with a Zilean than  a Cho'Gath for instance. I typically play AP Mid (preferring the Assassin sub-type) or Support (preferring the more aggressive supports). I'm also VERY comfortable with Malphite but drop off considerably after that.

So, for me, the two claring weaknesses on here are ADC and some kind of ganking jungler. I don't own Nocturne, Kha'Zix or Hecarim so I can't really get good with them yet (though it is Nocturne free week). Shaco, despite my love of assassins, isn't my cup of tea in the jungle. If and when they retweak Diana, I'll likely try to make her work in the jungle. I know her really well and her ganking ability out of the jungle would be pretty awesome. On the ADC front, frankly, Twitch is the only one I feel "comfortable" with. Corki and Ezreal have proven to be tricky in my hands. I don't own MF or Caitlynn and haven't tried Ashe.

Anyways, that's my homework for the next two weeks. Learn how to gank out of the jungle better while also improving at the two things I have an aptitude for.

--
It is better to be silent and thought a fool than to speak and remove all doubt
-Abraham Lincoln

The problem with Internet quotes is that you can never be certain of their accuracy
-Abraham Lincoln