Well, considering you're only playing Evelynn for fun (I have yet to make her truly effective against all enemies (stuns just drop her), despite her being one of my top 3)... I'm around mid elo with ~50 wins over loses (unfortunately, some loses were my fault because Eve -is- severely gimped if underfarmed, but that is true for all champions, but she's even more useless when underfarmed by harrassing champions, others are just because of team composition and feeders). Overall, Eve IS fun, but she's not the best champion to play if you're aiming to win
There's a jungling Eve guide somewhere on leaguecraft or the LoL main website that I couldn't find again, but after I read it, my Eve improved greatly.
Essentially, what I got from Eve was to fix her "weaknesses" - she's a stealthy character, and a fast one at that, so you want to be quick and efficient in your killing methods. However, she lacks the ability to effectively farm unless you have a character that can harass a lot, or you jungle.
Your summoner skills should be either teleport/ignite or smite/ignite. I prefer smite/ignite, because it allows me to farm while I roam around to kill unsuspecting champions that overextend. Generally, to be a good Eve, you have to be everywhere, while not having champions know where you are. By level 6, I usually can jungle VERY efficiently. Generally you want to be higher level than other characters, so jungling is very effective since your skills scale very nicely. The more you level stealth, the longer your stun is, and the damage on ravage is really nice.
For skills (jungling): 1 -> hate spike, solo golems/dogs in your lane, then continue jungling if you have the runes to do so, otherwise go lane 2 -> stealth 3 -> ravage (extra damage early game really helps with killing) 4 -> hate spike for more jungling/farming 5 -> ravage 6 -> ultimate Finish ravage, then either hate spike or stealth (depending on your playstyle, I'm usually fine with 20 second stealth, though having more just allows you to set up and stalk prey much easier... I prefer the extra damage for better jungling and killing).
Level ultimate according to your playstyle; it's useful, but it's situational, since yes, it does make you attack faster and move faster, but it's not necessary; the other skills help you farm/set up better.
------ My item choices usually are: Amplifying tome -> Lucky pick -> Deathfire late game (You want as much "burst" as possible... some squishies can be dropped from 100% -> ~30% and picked off from some hatespikes)
Boots of Mobility -> You want as much speed and mobility as possible! Chase down any champion and travel down the map asap to that one unsuspecting target. Having a lot of speed is simply very nice for Eve, specially when coupled with her ultimate, since you can actually walk into towers and kill low hp champions before the tower even hits you once!
Sheen -> lichbane OR mejai's(if you're doing well)
Guinsoo, voidstaff, or Zhonya's (it usually ends by here... and it's personal preference + what's needed endgame... Eve is amazing at pushing towers).
Yes, this is a squishy build, but if you're doing it right, and if your opponents aren't sticking together 24/7, you should do fine. Eve is really situational, and she loses quite a bit of power if she goes survivability. However, you can forgo some items for a sunfire cloak, since it gives you the much needed health and it helps with damage since you can burn their whole team while invisible.
Eve can usually kill anyone who is not a tank at ~50% (or less, situational) hp alone, if they aren't A) better than you or B) you're underleveled. Usually. If you can't kill them in one go, you're dead though. Most champions, as you stated, have higher burst than she does. But her initial stun and her burst usually lets her win majority of the time. --- I'm bored...