Alright, since there's a lot of commotion around napalms and drop rates recently, I'll put out my method which has so far proven more than efficient. So listen up.
-First of all, we need to take a look at the elixirs/enhancers we will be using because, let's face it, without them it's like trying to catch a cloud with your bare hands. Note that a 4x enhancer/elixir is NOT necessary for this, the 4x damage does speed things up though. So the options are: 1. 4x Enhancer(1 hour) for 75 platinum. 2. Miner's Luck(30 mins) & Speed Elixir(30 mins) for 25 platinum in total. You may want to use 5 more platinum to add a 2x damage elixir(30 minutes) for maximum efficiency.
-Secondly, if you're serious about wanting to farm a napalm, don't expect to do so using your 50-51 character. There's a game mechanism called Threat Level, which I've analyzed in a previous guide(that got taken down because it contained most of the glitches/bugs I've found so far and because "the game wasn't meant to be played this way"). Anyway, I'll add another topic on it later.
For now, all you need to know is that you will want to farm using a level 33-36. Player class plays no much role. The minimum level to enter hive is 33. What we're looking for is called Red Threat level. It's the best you can get and it offers a significant experience multiplier boost as well as greatly improved drop rates. For the Red Threat level to be active, you'll need to have at least 5 level difference(means your player should be 5 levels lower) from the maximum level displayed on the level limit of each campaign(eg. In this case, Hive has 36-41).
In addition, you will want to freeze your experience gain from the Social Options(Options>Social Options>Disable Exp Gain) when you reach level 36, because leveling it higher messed the Threat Level. I find it very efficient to have this second character on the same email/account so you can switch to it when the time comes. Read on and you'll understand.
-Lastly, here is the strategy/tactic. To avoid wasting valuable minutes from your enhancers, it's best to finish 90% of the map on another(high leveled, preferrably) character. Then switching to your enhanced alt just for the Queen drop. And repeat. This method has proven to be the most efficient, both platinum and time wise. As a checkpoint for the switch, I'd suggest after clearing the room before the boss area; the one with the 5 blue large vulars with the insanse DPS.
But what matters to get the timing for the switch and not end up running half the map to find that the others have killed the Queen before you, is who you're running with. If it's another 2-3 level 50-51, you'd best switch a little before the room I mentioned. If you're going solo, you might as well clear the boss room and get the Queen to 5% before you switch. And by solo, I mean a locked game which only you can enter.
I'll have you know, when you quit a game it doesn't instantly disappear from the face of the earth. If you search for it in the Sector Map, you'll see it for around 30 seconds with 0 players. So a fast switch is necessary if you're going solo. Although solo is not advised unless you really hate everyone because full team runs are much much faster.
Well, there you have it. Be patient and you should see results in this method much faster than in any other. Just a footnote, this method doesn't guarantee you'll get a napalm, I just calculated the best conditions for a pink/legendary rarity item drop. You can thank me later.
Any questions welcomed. Have fun!