Basically, for the "Dubious/Taunting/Roar/Rise" Challenge achievements, you have to progress through the challenge levels. Self-explanatory, just choose whichever challenges you find easiest (that meet the requirements) as you go through.

I attached an image of the unlocked achievements with their descriptions, which should give you a good idea on what to do for each.



"Start//Finish", "Pulling Teeth", and "Rookie Mistake" don't need any further explanation, you just do as the description says, and you're also most likely to get these normally as you continue playing through the mod. "Skull Storm" is also self-explanatory, but incredibly difficult. I recommend becoming familiar with a starter deck and picking a strategy that you think will carry you through the game. Try to experiment outside your comfort zone if you're really struggling though, it might help.

For "Master Trinketeer" you need to solve the oil painting puzzle (and take the item!) in three separate runs for the achievement to unlock. If you're having trouble getting the required cards for the puzzle, I recommend the "DeckEditor" mod by Peaiece, or just praying to the luck gods.

For "Bad Moon", the easiest way to do this is with some Magickal Bleach (which will erase all the sigils from the moon, and this also works for the Limoncello during the pirate boss) which you unlock from the oil painting, and flying cards. Having the fan item is also very helpful as combined with the Magickal Bleach you can most likely have your cards deal enough damage that turn to K.O. Leshy.