I will explain the walkthrough based on how to do it on the highest difficulty level, which you cannot unlock until you finish normal difficulty. If you play through the game this way, then you should have little trouble when you get to the hardest difficulty.
Here is a short breakdown on the differences between the difficulties.
You can die as much as you like. All money and exp that you have earned each life will carry over. There is plenty of food in the chests, and the mobs don't hit too hard.
Mobs spawn more frequently and hit harder. When you die, you lose all progress that you have made in that life. You can retry the level based on the items, money, and exp that you had upon completing the previous level. If you are on level 1 and die, then you lose everything. Food is still plentiful.
Mobs spawn a lot more frequently and hit MUCH harder. Most mobs will kill you in about 4 hits. There is NO FOOD on any of the levels. You will be healed to full at the start of each level. If you die, you will lose all progress you have made on that life in the current level. You can replay the level based on the items you had from the previous level. If you were on level 1, then you will lose all progress.
The first skill to level will be Agility. This will allow you to run faster and swing your weapon faster. Take this skill to 5 before putting points into anything else.
The second skill to focus will be Endurance. Take this skill to 10 before spending anymore points. This will allow you to take less damage from hits.
The last skill is strength. If you are on easy or normal difficulty, you can take this to 10 after you have capped out the previous 2 skills. If you are playing on hard, you probably won't get enough points to level this up.
Money and Chests:
It is extremely important that you open every single chest from level 1 to level 4. By the time you reach the end of level 3, you should have enough money to purchase the sword. The sword is an extremely powerful weapon and will dispatch most enemies, even on hard with only a few hits. When playing on hard difficulty after you have enough money to purchase the sword, you can skip the rest of the chests, as they only drop money, and you won't have a need for it any longer.
Jumping Tip: When doing the jumping puzzles, be sure to aim your camera downward so that you can jump at the last possible point to ensure that you make the jump. I suggest a downward angle of about 45 degrees.
Below, I will add videos for each mob and for each jumping puzzle to acquire keys. The dungeons seem to be randomly designed each time you play them so including a map isn't really an option. The hardest part about the game is understanding how to do the jumping puzzle on Level 4.
I have included the achievements below in the order in which they are done. The achievements are not retroactive, so you will have to complete the game 3 times at once for each difficulty in order to get all the achievements.
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