Hyper Light Drifter

New Game + is unlocked after beating the game once, and can be accessed from the main menu. It doesn't matter if you overwrite an old save or make a new one.

In this mode, you skip the tutorial and start with all abilities. The only things you won't have are weapons, bombs and healing upgrades, but all those can be bought from the stores as usual. What makes NG+ hard, however, is that you only have a maximum of 2 hit points. This means that any damage will put you into critical health, and many enemies can outright instakill you. That may sound horrible at first, but most of the game is really still quite manageable even under these dire circumstances.

Since you only have to beat the game and you start with most upgrades, you'll want to ignore everything except the bare minimum required for completion; 4 module pieces per area, and raising all the pillars. That said, I highly recommend getting enough gearbits to upgrade the ammo capacity for whatever your main gun is (I prefer the standard pistol) and buying the healing upgrades. Most of the time you won't have a chance to heal since you die so easily, but when fighting weaker enemies, more healing really comes in handy.

Here's what my route through the game looked like. You can use this map as reference for where the modules are:

North:
First module you come across on the invisible path on the left, easy free one.
Second module just up to the right after teleporter.
Take the lower of the two elevators on the left side, get the first of the two modules in it.
Last module is on the path to the boss, hidden passage in a room where you have to kill all enemies to proceed.

Boss: This guy is by far the hardest fight in NG+. All of his magic square attacks are instant kills, so you really have to learn how to dodge them.
Phase 1 (100 - 75%HP): He moves to a corner, summons a "snake", then either an X or + shaped set of squares on your location, and then another one. Hit him twice as soon as he appears, then 2-3 times in between the two X/+s. For the snakes in both his normal phases, it's always safe to hang out in the opposite corner, they don't reach that far. After a few repeats of this, he moves to the center. When he does, immediately run in and press X, X, X(hold, release) to do two quick hits and a power attack. After this, retreat and shoot at the boss and the birds he'll have spawned. The birds only do 1 point of damage to you, but you definitely want to kill them all right away.
Phase 2 (75 - 40% HP): A straight upgrade to the last phase. The Xs and +s are bigger and he summons two snakes at a time instead of one, but I actually find it easier than phase 1. Same tactics apply.
Phase 3 (40 - 0% HP): The lower you can have his health when he enters this phase, the better. He'll start moving from corner to corner, summoning rows and columns of death that move across the room. This is what makes the fight really hard. Follow the boss closely and continuously deal as much damage as possible, but pay VERY close attention when the magic lines are moving towards you to make sure you're in the right spot. A location that's safe on one pass of horizontal lines may not be safe the next time, as they switch it up sometimes. When the boss only has a few hit points left, he moves to the center to summon birds, which leaves a good opportunity for you to land the finishing blows.

East:
First module is free, in the underground part on the way to the teleporter.
Second module is pretty much unmissable after you take the north path in the southeastern underground part. Yes, the big enemies instakill you, so be careful with them.
For the third module, enter the left door below the teleporter, and pass through the first two rooms. There's a module in the third room, but you'll have to survive a pretty tough fight to get it. Abuse shot reflection and focus on taking out ranged enemies first is all the advice I can really give for it. Bombs may also help if you have them.
Fourth module is free when you come outside after the boss.

Boss: Speaking of which. The frog is a joke compared to the bird, but he still packs a punch. Everything he does kills you instantly, even just having him land on you, so be very careful of that. The key to this fight is making the little enemies he pukes out explode close to him to knock him out so you can deal easy damage, but that's sort of luck based since he moves around so much. It's nice when it happens, but if you can't manage that, you can hit him twice quickly over and over when he's throwing jars (dash away so he doesn't smash you), then shoot him once your ammo is full. Dodge during his jumping phase, and repeat. He doesn't get that much harder as his health depletes, he just kinda throws more jars.

West:
First module is once again free, in the hidden underground room to the south before you're even in the area proper.
Second module I got is in the southeast of the room before the teleporter. The swordsmen surprisingly only deal 1 damage, but don't underestimate them. Take out the wolves first when those bastards are around, preferable with two pistol shots.
Third module is southwest of the teleporter room, through a hidden passage and an invisible bridge. This area is full of wolves, but if you play it safe, they're pretty easy to deal with as they often get stuck on scenery.
If you take the north route from the teleporter as I did, the fourth module will be in the northern of the three underground rooms. After this, you can dash past pretty much every enemy up until the boss, and I recommend this course of action.

Boss:
This guy... not as hard as birdman still, but definitely a step up from frogman. When he lunges at you, his first attack does 1 damage, but the follow-up strike is an instakill. Make sure you learn the timing to dodge both. Try to make him break as many of the crystals around the room as possible as soon as the fight starts, as that means you'll have to deal with less enemy reinforcements later on. After the boss attacks, you can safely hit him twice before getting ready to dodge again. After three attacks, he'll stop and do an instakill AoE attack that shoots out icicles that will also kill you. Dodge this and he'll shoot three balls at you, the third one slightly delayed. Can't remember if these do 1 or 2 damage, but you should reflect them at the boss either way. After one cycle of all this, he might start summoning enemies from the crystals that are still intact. Kill these enemies if you can, but you don't have to. At about 50% health, the boss will do three attacks in his chain instead of two, so dodge accordingly. It's a pretty chaotic fight, but keeping your cool is the key to winning. Never just mash dash to dodge, always time it properly and always dodge AWAY from the boss. This isn't Dark Souls.

South:
I went for the scythe wielding miniboss for my first module piece, which is accessed through the eastern elevator and then following the rooms south. Dodge his attack when he runs at you, hit him a few times, dodge whatever he does after that (he either throws the scythe at you or spins it around himself). He's kinda hard to hit with the gun, but overall he's not too bad.
Use the east elevator again, and take the east path underground. Both the second and third module pieces will be in rooms on the way to the "real" boss of the area, check the map to see which rooms.
My choice for the fourth module piece can only be accessed once you've raised the pillar in the area. It's the southwestern entry point to the underground, and you'll have to use an invisible bridge to reach it. A simple dashing exercise is all it takes to claim the module.

Boss: This is a very hectic fight. The boss' arrows only do 1 damage, but the mines kill you, and what's worse is that if you die, you have to pass through the room BEFORE the boss again, every time. I just got used to dashing through it without fighting, but do as you please.
For the actual fight, this is all about aggression. The longer it goes on, the more mines will fill the floor, and trust me when I say those suckers WILL eventually all explode at once. If you're in a safe spot, this can actually end up dealing massive damage and even killing the boss, but don't count on it. The biggest tip I can give is to make sure you dash through the arrows the boss shoots. They count as projectiles, and absorbing them will recharge your ammo very quickly. Just don't dash into any of the mines, and try to deal damage as quickly as you can to end the fight before it gets bad.

Finally, the final boss. This guy is actually a complete pushover once you know his tricks. He'll dash at you, just dodge to the side. He'll move to a corner and shoot some stuff at you, just reflect it back, it only does 1 damage even if you take a hit. Look to see which way he's gonna shoot his beam, and move either above him or next to him to avoid it and get a great chance to deal massive damage. When you've done enough, the screen will go black, and four little things will appear around you. Destroy at least one of them, and stay far away from the others. Avoid the boss as he appears, then hit him as much as you can. After this, the cycle repeats, but the boss will dash at you twice in a row. Next time, he does it three times, but other than that nothing much changes. Kind of a breather after some of the other stuff you've been through.

If you have any questions, extra tips or need advice about any specific parts I've forgot to mention, please let me know in the comments. Good luck!