In order to unlock the Samurai difficulty, I believe the game must be completed on a different, lower difficulty first. I would recommend playing through on Normal first (to get that achievement).
Before starting on the Samurai playthrough, I would recommend to do a few things.
-Collect every Psycho Pick possible. There are three Psycho Picks exclusive to the Samurai difficulty so upgrading everything won't be an option until the end of your Samurai playthrough.
-Collect both of the Amulets that allow co-op.
-Buy as many Nutribreads and Uberfish as possible. If you need swirls/gold, feel free to replay "Cam Core" on Easy. This will net you ~100 swirls/gold per time, and it's a short level.
For the upgrades, my character was like this:
-Shift Sword: Maxed (recommended)
-Chainsaw: 2 Stars
-Arsenal: 2 Stars
-Health: Maxed (recommended)
-Magic: 4 Stars
Feel free to upgrade as you see if, but Shift Sword and Health should be maxed out before playing on Samurai. Now it's just about time to actually play through the game on this difficulty. It should be noted that enemies are a lot harder this time around. Foes that normally appear at the end of the game will come much earlier. Sometimes bosses will make an appearance as regular foes as well.
As everyone else has mentioned in their guides, it's highly recommended that the game be played on co-op. However, I would actually say to play the game on co-op with another person, instead of trying to play as two characters yourself. Let's break down the roles.
Player 1: The top priority for Player 1 is to dodge anything and everything possible. To do this, Player 1 will always have the Shift Sword equipped and will be constantly teleporting in battles. While it is very possible to use a rubber band to do this, I strongly advise a human to control the dodging. Sometimes, you'll get stuck in a corner and get attacked by foes that'll shoot you. The attacks the enemies dish out are vicious; they kill in 2-5 hits.
So, Player 1 needs to dodge, dodge, and dodge. Attacking is only recommended when your partner is alive and there are very few enemies. If there's a large boss and you have magic, feel free to use it and then teleport away. What you have to look out for, mostly, are the Agents (shoot up at 45 degree angles), Assassins (they have an air lunge attack), Freaks (jump straight into the air and can seriously hurt you), and any robot with a homing missle attack (can take a 1/3 of your health away).
Player 2: If Player 1 has to strictly dodge, Player 2 has to strictly attack. There are a few disadvantages to being Player 2. Player 2 will never have the same amount of health as Player 1. Player 2 will also never respawn with full health. However, the main advantage to being Player 2 is the infinite respawns. Player 1 will have to use continue hearts when they die, but Player 2 can just respawn forever. Granted, there's a 5 second respawn time, but that's not an issue. Player 2 should also have a grasp on how the game plays as well as what combos work the best. Player 2 will always respawn with whatever weapons Player 1 has equipped. So, if Player 1 has a Shift Sword out and the Kamas as a shortcut, Player 2 will have the same when they spawn. This is helpful to know when Player 2 wants a different weapon besides the Shift Sword.
Player 3: This one is completely optional, but if you have a friend willing to play the game with a Guitar controller feel free to let them come in and play. The guitar character will always float near Player 1, acting like a guardian of sorts. While the damage the guitar does is quite minimal, I have noticed that it'll cause regular enemies to flinch, allowing for Player 2 to dish out some quick combos. This can be helpful for keeping Player 2 alive for a while longer.
General tips for Samurai:
-Always try using the appropriate kill on enemies. This will net more swirls, grant health, and sometimes magic. Just be sure you're close enough to get them.
-Player 2 can use whatever items Player 1 has purchased, but that isn't really cool to steal healing items from Player 1.
-Grabbing and slamming an enemy into the ground from a high point will net a small amount of HP. This is helpful for trying to keep everyone alive.
-At the start of a new level, feel free to quit and start again. This grants full magic to Player 1 (and full health for both players).
-Player 3 is invincible.
-The Shift Sword has some pretty decent Y attacks that help damage foes and keep you moving on the battlefield.
-I find that using the Arsenal on enemies that jump around a lot can be useful. Just juggle the enemies with an upward X attack over and over (this is particularly useful on the Invisibles).
-If Player 1 gets killed near the beginning of the level, don't waste part of a continue heart. Just start from the beginning again.
-There is a spot in the level where you fight Chef, I believe, that causes the player to glitch out of the level. I have experienced this three times so far. It's near the beginning of the level, where Steel Magic is unlocked. If Player 1 teleports near the walls, it's very possible to teleport over them and fall off the level. There is no way out of it other than to restart the level.
It's been a while since I played the game so I might have made some overly vague statements, but if anyone has questions or comments, feel free to let me know.