I know in the past I've said that which champion you play doesn't really matter if you just have familiarity. I still think that's the most important thing because winning early in League usually means winning the game. However, that doesn't mean that composition isn't important. For us scrubby low tiers, what I mean by this is not that you have the perfect synergistic composition. Instead, I want to talk about analyzing what composition you do have and what you can do with it.
First, let's look at combos. If you have any obvious combos, you should consider working with those options. Likewise, if your enemy team has combos, you should be looking to play around those things in teamfights. It's more important to think about the enemy's combos than your own, honestly. Most combos in LoL are obvious. For instance, you might have Jax and Lulu, and you might think of Jax leaping in with Counter-Strike, then Lulu ulting him and getting a big AoE wombo from that. Or you might see a Veigar stun and link it to any other skillshots in your team such as Ahri charms, Nidalee spears, etc. Likewise you need to be aware of those things on the enemy and play around them.
Be aware of big ultimates or even normal skill CDs and try to fight when they are down. If the enemy team makes a failed play on one of your characters and burns an Annie ult on it, you need to engage shortly after Tibbers times out (or when you kill him) with your own fight-starting ultimates.
That brings me to another point. What if you don't have initiation? It happens. This means you can't start fights, really. If you can't initiate, don't try to start a fight if the enemy team does have initiation. Just wait for the enemy team to start something so you can deny them a good opening, then use your team's advantages to take the win.
What are your team's advantages? You might need to kite the enemy through traps, Nidalee spears or Teemo mushrooms before you fight. Don't assume that you can just go in. You might need to pick a single enemy (with a Blitz grab or something) before you can fight.
I'm poorly communicating the obvious, here. When you get into a match, you should always look at what your team can do and formulate a strategy. Don't look at individuals and don't skip characters. I talk with friends and they always think about what their character can do in a match, and I'm like "well we have xyz, which works together like this." Also compare it to the enemy team. Don't just look at one thing. "We have this OP thing" but you don't have anything to work with it.
This isn't about drafting teams; it's about playing with what you have. It might also be the case that you actually draft a champion to fit your team comp, but again, the point is to be flexible and play with what you have. You'll win a lot more in the lategame if you play to your team's strengths, and at lower levels of play, the lategame might actually be where the game is won.