Army of Ruin

These "minimal upgrades" challenges are pretty tough and stressful. You should definitely wait to do them until you've unlocked more/better weapons and trinkets and characters, because you can still choose from all of those during a run. I used Selina for all of them just because her skill means you don't have to rely on constantly trying to find food drops, because you will be taking plenty of damage. Below, first, are what I spent the allowable funds on. Feel free to change it up, but I did all three challenges first try. 

Forge I - no upgrades
This took me quite a while to defeat the final boss, and I have no idea how I managed it considering I went multiple minutes slowly whittling its health down. 

Forest II - 5,000 allowable. I spent the following:

• Damage - 4
• Experience - 3
• Regeneration - 1

I did it first try, but it was stressful keeping enemies away. I got Crimson Flask which helped tremendously, especially while trying to defeat the final boss. 

Graveyard III - 20,000 allowable. I spent the following:

• Damage - 8
• Experience - 4
• Regeneration - 2
• Favorites - 1

I know, the Favorite makes no sense, but I Favorited Forbidden Sword. I also got very lucky and got Forbidden Shield too (+75% fire rate), and even with BOTH of those maxed, I was still barely controlling the horde. I had Ranger's Bow, so I spent the majority of the stage just walking to the right, behind my Bow shots, since that was doing all the knockback.

I found them to get progressively easier in the order listed above (Forge the hardest, Graveyard III the easiest). There isn't a whole lot to give in terms of recommendations, because you're entirely at the mercy of RNG for these runs, and you are unlikely to be able to evolve more than one or two weapons just because you only have four slots available, and what you get is what you get. 

My advice is to try to balance types of attacks. For example, you don't want only attacks that surround you and do AOE, of you'll find too many enemies moving in around you to keep off. You also don't want only weapons that shoot in set directions (like Trident, Spear, Spectral Dagger) or you'll have a tough time too. I tried to focus on a good mix of weapon types, and I didn't worry about trying to sync trinket elements with weapons. It's better to have helpful trinkets than have the "correct" ones but all they do is increase move speed or experience gain or money, none of which helps you survive. I highly recommend getting something that adds knockback as a high priority. You've very unlikely to be able to kill enemies quickly, so the best way to control the hordes are with some sort of knockback (Crimson Flask and Silver Bullet being extremely useful). You'll also almost assuredly need something that increases damage. Wristband of Grudge is really good, because you'll be getting hit plenty and it adds a ton of extra damage when you get hit. As things progress though, killing enemies isn't a priority. Keeping them off you is the priority, and you're just biding time until the boss shows up. Then, you can carefully whittle down the health of the boss and don't do anything reckless, and you should be able to manage it.