First of all I would like to say that I did not test this guide myself yet, as I had no volunteers to join with me.

But the bottom line is Evelynn and Vayne jungle, Veigar mid, Galio bottom and Zilean top.
I will only go in on the jungling and teamrelated subjects. How to solo lane with veigar, zilean or galio is something that you are required to be able to for this to work.

Evelynn and Vayne start with buying a Cloth Armor, 2 Health Potion and a Sight Ward.
Now you might ask yourself: why not 5 Health Potion? Why 2 Sight Wards?

Well here is where the tactic to succesful double jungling comes in already.
There are a few possibilities:
1: They have a jungler starting at Red buff
2: They have a jungler starting at Blue buff
3: They have a mid/jungler who needs blue buff + they get help from allies
4: They have no jungler at all
5: They have a counter jungler

So what to do?
Both rush to their blue buff.
Stay in the bush near their first bottom turret (so not the bush near blue buff itself)
Evelynn will now go stealth and check up on their blue buff.
1: They have a jungler but he isn't at blue buff. Put a Sight Ward in the bush of the blue buff. Kill the blue buff fast together. Double Smite him from ~850 hp. Immediately go back to the bush near the outer bottom turret from your opponent and teleport to your inner mid tower.
2: Evelynn will be in stealth near him while he kills the blue golemn. If he has very low health stun and kill him. Vayne rushes in from the other bush and nails him to the wall for double stun. This way you didn't use your smite to get the first blue buff. If he has a lot of defense and doesn't get a lot of damage from blue buff then kill steal him. Make sure Evelynn stuns him and then double smite the golemn (so only go when you are sure that you can double smite the golemn) Put down a Sight Ward in the bush also while he is stunned. Now make a run for it and teleport to your inner mid tower once you are safe.
3: This is false hope. Against 2/3 champions you can do nothing safely. Not even with 2x stun. If this is the case, just go back to your own blue buff.
4: This is easy. You kill the blue buff golemn with the two stun abilities and double Smite him at ~850hp. Now you take the wolf camp and you teleport back to your inner mid tower.
5: This means you do the same as on 1:. You kill the blue buff golemn, put a Sight Ward and teleport back to the inner mid tower.

What is next?!
Well now you've probably got one blue buff and you warded their blue buff. If they have a jungler that started at red buff he will be astonished at how fast you took his blue buff and he will most probably fail his route, or at least lvling sequence.

At this point you are with one buff at the inner mid tower. You now kill the wolf camp and continue to your own blue buff. This should go easier now due to the fact that one of both champions has the blue buff already. After this you will both have it and you go to the wraiths. Kill them and go to the red buff. Let vayne take the red buff (very important) and take the 2 golemns at bottom.

Now you both go mid to gank. This really can't go wrong if you ever played Evelynn or Vayne before. And in combination with Veigar it's even easier. Evelynn goes stealth and stuns the mid from behind. Vayne hits him once with auto attack to slow him with the red buff and then uses Condemn to push the opponent against Evelynn. This will stun him again. In the meanwhile Veigar and Evelynn keep hitting the enemy. Finally if he is not dead yet and wants to run Veigar places a stunblock in him so to stun him once more. And by now even the strongest mid should be dead.
(MAKE SURE YOU STUN AS FAST AS POSSIBLE AFTER EACHOTHER, TO AVOID YOUR ENEMY TO USE FLASH TO GET AWAY!)

From here you split up for a while. Your jungling phase was so quick that you ran out of camps, unless your opponents have no jungler and you can take their red buff. Generally I think it is better to split up. It is important that Vayne goes bot and Evelynn goes top. I'll come to the "why?!" in a sec.

Evelynn goes top, possibly ganking one of the opponents and otherwise just taking over the lane if Zileans needs to go back. If not just farm minions untill you are lvl 5 or 6. By now you should have enough money to buy Boots and probably also for Sheen. If not just go with the boots and an Amplifying Tome.

Vayne goes bottom, and I suggest to gank one of the opponents since you still have the red buff to slow them. This makes it quite easy to get a kill in combination with the slow from Galio. Now farm untill you reach lvl 5 or 6. At this point you also return to base and buy Boots and Emblem of Vaylor.

Now you can go to the dragon inmediately or you get the blue buffs first again (but be careful for junglers and counterjunglers!) You can teleport to your Sight Ward which is at their blue buff to get there quicker. If they have a jungler, you should not be afraid that he took the dragon already, since you need around lvl 7 for it at least. Most junglers only attempt to take it at lvl 8 or 9 and solo jungling goes a lot slower than dual jungling; especially if you stole their blue buff at start. So now you either walk to the dragon or teleport to any nearby ally/ward/minion. Take the dragon on lvl 5 (Yes it is possible, but make sure that you defend one by one and stun a lot. Also double Smite the dragon at about 1k health.)

From here on you go back to your own jungle and go the regular path. I suggest to go back and stock up on some items first, but if you have enough health you can go on by doing: blue buff > wolf camp > wraith camp > red buff > golemns

After this you can go back to buy Sheen if you had not gotten it already or some other part of the build. Or you move on:
From the golemns you go to their blue buff again > wolf camp > teleport to top minion or tower > gank their top > golemns > red buff > wraiths > gank mid.

Tip: let Evelynn always go in stealth. If the enemy has any Sight Ward they will only see Vayne passing by. If they attempt to gank Vayne they will be in for a small surprise when they suddenly get stunned and backstabbed by Evelynn. After a while they will notice this and put down Magical Sight Ward. This is not really a problem. They will spend yet more money on wards and the only thing they accomplish is seeing Evelynn once in a while. You are still together, so whether they see you or not you still both have stun and most probably a good feed, making you do a good deal of damage.

Well so far I guess this is it about jungling. Now you will both be strong enough to take one jungle if you want to continue jungling. You can go roaming or laning also now. Evelynn should take the top jungle as to avoid the tower damage moving from the wolf camp to the wraith camp or back. If they have a jungler I strongly disadvice you to go on jungling, at least not solo.