The Oni TV Show is a side quest that starts you in an endless mode with your default gear.
I would like to point out early that this achievement will lead you to unlocking gear that will be required to unlock future collectible achievements.
Each wave is randomly generated and may contain enemies from any of the game's levels. Due to the random nature of the mode, some waves may be extremely tough.
To help you out, after each 3 waves, you are awarded a choice of weapon, projectile, amulet or stone to help out your run. There are three choices each time, and you are allowed to choose one. As you may expect, these are also completely randomized.
There is loads of luck involved in this mode, but I have several tips that should be able to help out:
Play through the game first: In addition to getting used to the game and honing your ninja skills, you will also encounter all the possible enemies and learn the tactics to defeat them.
Learn your equipment: The toughest part about making a informed decision about what gear to pick up when offered is to figure out what it does. Unfortunately, you are shown the item, but are not given a name, stats or any description. You must be able to recognize the item and have an idea on what it does. If you play the game early enough, most of the gear you are offered will be new, and you may not have any clue as to what it actually does. Once you have played through the game, you should become familiar with most of the weapons, projectiles and amulets to the point where you can strategically choose effective gear.
Upgrade your sword: This is obvious, but it will be one of the first things you will want to upgrade. Doing more damage will make surviving waves much easier. Unless there is something far more useful (or none of the swords are worth it) you should upgrade your sword as soon as possible. Try to determine if you sword is elemental. If you get an elemental sword and no stamina boosters, you may find yourself unable to attack if you lose your stamina.
Use good use of amulets: In my opinion, you should focus on any amulet that helps with your Health. Cursed skulls, health boosters or the invincibility shield are good choices. You don't get to choose which slot your amulet goes. It works on a 'First In, First Out' basis. In other words, if you pick up a 4th Amulet, you will lose the first one you picked up.
Pick up an elemental stone if you can: In addition to other equipment, you will sometimes see one of the three elemental stones (fire, ice or dark). Choosing this is helpful, but only if it fits your strategy. For example, if you are using a knife weapon that allows for faster generation of your special meter, or you have an amulet or two that boosts your special, then the stones may be a good idea. If you are finding yourself heavily relying on projectiles (which don't add to your special meter) then you may want to pass.
You don't need to select anything: If you are happy with your setup, you can opt to not choose anything. After several seconds, the items will vanish and a new wave will start. I actually had everything I needed by wave 20 and just let every other selection time out. Like I said, heavily luck based.
To give you some insight on a decent (albeit luck based) strategy, I will share my setup. As I mentioned earlier, I got this setup earlier and opted not to pick up anything new after wave 20. I actually got into the 60s and decided to let myself get killed off at one point.
Sword: Arcane Knife (ok damage, but quick attacks helped fill up special meter. Also, it is not elemental so I didn't lose any stamina using it)
Projectile: Sharp Crystals (very weak but great for setting off the explosive enemies. Also, it made racking up a quick combo easy for one of my amulets)
Stone: Fire (the element isn't that important, just happened to be the one I saw available)
Amulet 1: Cursed Skull ++ (gives you 1 hit point in health for each enemy defeated).
Amulet 2: Cursed Skull ++ (turns out, this game mode allows for duplicate items)
Amulet 3: Cursed Skull + (This one gives you one HP for each enemy defeated when you have a combo of 15 or greater. This is why having the knife and sharp crystals were beneficial).
Basically, I was getting 2-3 HP for each enemy I killed. In addition I was getting bonus HP each wave for high combos. Even more if you don't get hit in a wave.
You may not be lucky enough to find all of this in a single playthrough, but it should at least give you a good understanding for an effective strategy.