I was taking a look through a Neo Discovery spoiler...and I saw this Beedrill.  I constructed a deck on Apprentice and have done pretty well so far.  Here is the deck list:
Pokémon - 12 Basic/ 2 Stage 1/ 3 Stage 2
3 Weedle (Neo II)
2 Kakuna (Base)
3 Beedrill (Neo II)
3 Cleffa (Neo)
3 Electabuzz (Base)
2 Rocket's Zapdos
1 Brock's Mankey (Taunt)
Trainers - 29
3 Professor Oak
3 Computer Search
4 Item Finder
3 Pokémon Trader
3 Pokémon Breeder
3 Lass
3 Nightly Garbage Run
3 Gold Berry
2 Scoop Up
2 Rocket's Training Gym
Energy - 14
7 Grass
7 Lightning
The basic strategy is to use the Cleffa/Lass combo first turn (Scoop Up helps here...if I start with something other than Cleffa I can scoop up my active to promo cleffa) and retreat for a Beedrill by turn 2.  I then use Triple Poison combined with Rocket's Training Gym to make it difficult to retreat (there's nothing worse than finally poisoning a Sneasel, only to have it retreat for another Sneasel then come hit you again).  Nightly Garbage Run helps get my evos back and also helps with the low energy count.  Electabuzz is helpful to stall if I'm caught without an offensive force, and it can also take a few prizes while I'm preparing another Beedrill.  Zapdos is used as my cleaner in the late game when I can't seem to get another Beedrill out, and he's also my main weapon against Fire decks.  I chose Brock's Mankey instead of Magby or Cleffa as my anti-Slowking tech since I don't have any Gust of Winds in the deck, and Brock's Mankey is more useful in taking out other support Pokemon that do not rely on powers.  I think everything else is self-explanatory.  Hope you enjoy the opportunity to fix a somewhat original deck.
I looked through my inbox to find something that doesn't include Sneasel or Murkrow, since it seems like almost every fix I do uses them. The deck I ended up on is actually really good, it'll be hard to make it much better. But nonetheless, I'll see what I can do.
I suggest you use the Neo Discovery Kakuna rather than the Base, it's better in my opinion. Let's add in one more Weedle too just to make it easier to get Beedrill out as fast as you can. Everything else looks good. I like that Brock's Mankey in there, looks like it would combo well with your Rocket's Training Gym.
With the trainers, let's start off by making some room by dropping two Item Finder, a Nightly Garbage Run, a Pokemon Trader, and a Scoop Up. You should be able to do well enough with one less of those cards. In their place, add in another Oak and Computer Search, and another Rocket's Training Gym could help when you face a deck with a lot of free retreaters. Then add one Gust of Wind, for when you can't get Brock's Mankey out or you don't want to waste an attack bringing out one of their benched pokemon.
The rest of your trainers and your energy look nice. I hope what little help I could offer was of use to you. Good luck with the deck.
~ Monkey