First blood. Get the lizard buff with the help of your lane partner and go for first blood by level 3 at latest. Depending on your laning duo, level 1 FB is quite possible. Remember, Ignite first so you get +10 AP for your ravage and hate spike spam.
You want to angle yourself so you're behind the enemy and to their side, at a diagonal while also being further than 100 units away from a minion, so the enemy champion will be getting hit by your hate spike for full damage.
Don't be afraid to switch targets if the enemy's lane partner is throwing themselves at you to save your original target. You don't really care who dies, since the goal is to farm both of them for as much gold as possible.
You need to set a gold gap precedent that will snowball for the rest of the game. Once you hit level 5-6, you should cease laning and start cross-lane ganks with your team. This will give your former lane partner an xp and gold boost.
The main reason you needed to get early FB and ganks is so your former partner isn't 1v2'ing his lane against equal leveled enemies.
AP Evelyn is an anti-carry. You are not a real carry. You cannot tower push very well. Your role is to keep the enemy carries gold-starved and dead more often than alive. Much like a tank/support, you will pave the road for your carry to faceroll the enemy with superior gear and levels.
The more proficient your ganking potential becomes, the closer the laning era comes to an end, and the game will devolve into 5v5 roaming gank squads. Your team should form likewise and start pushing.
As AP Evelyn, properly geared. You will not initiate fights. The enemy can FF you and is probably already conditioned to do so by now. You will have your team's tank initiate, and as the opposing carries engage, you will quickly Ravage > Hate Spike them to death, and run away.
Why run from a team fight? You disposed of their primary dps source. Second, 1 or 2 of their team will inevitably change focus upon you, giving your team an additional few seconds of free damage.
In the 2 seconds you backed away, assess the situation. Whose team is winning? Are there low hp champions on the enemy side? If the answer to either is yes, return and assassinate said champions. You are like a seasoned surgeon cutting a person's vitals with a hot scalpel. You are a deadly and precise and a force to be reckoned with.
Keep note of who on the enemy team has oracles. It will usually be the ranged dps/carry. He is the first to die in any team fight, usually.