There's no way around this one, you have to reach level 20 to unlock it. The best way to mitigate the grind is to use the Mutant. A flat +200% rate will get you there much faster

The Cryptid is unlocked by finishing a wave with ten or more living trees. The Explorer's stat of increased trees spawning is a good way to help you along, especially if you find the item with the same effect, but this is a double-edged sword. The Explorer also starts with a lumberjack shirt, meaning you'll want to avoid turrets, explosives, any sort of crowd control, really. The last thing you want is for an explosion to destroy your last tree right as the round ends.

The Fisherman isn't complicated to unlock, but it does require a little luck -- you just have to get two baits in one run. As far as getting both of them, the Unarmed character has a higher chance of finding items that affect damage, including bait, but surviving the onslaught with them will be tricky. Jack, on the other hand, trivializes surviving them since he has fewer enemies per wave, but then it's up to luck if he gets the items.

This one is very easy. You have to end a wave on one hp, which might make you think of the ghost item that starts you at one hp. But you don't need to muck around with that when you can just use the Pacifist, walk into nine enemies on the first wave, and avoid them for the rest of the wave. Ridiculously simple.

The King is unlocked by having 3 tier IV weapons at once. Your best bet there is to buff up your Luck stat, especially if you can be fortunate enough to get an Anvil. The Lucky class is a good place to start. And again, seriously, pray you get the Anvil early, as it will trivialize this one.

You have to have ten tier one items in a run to unlock the Renegade. My first thought was that this had to be tier-one weapons, where you'd be better off using the Wilding so you don't get anything better than tier 2, or the Multitasker so you can have twelve weapons at once right off the bat. Those are still valid strategies -- Multtasker in particular will probably be very fast -- but most runs with any sort of Luck investment will get this no matter what because most of the items you find in crates will be low-tier. This isn't the fastest one to unlock, but its by no means difficult.