- Legitimately

Physical

Currently, the Complete Adventure is the only disc that will work physically. Attempting to download any episodes will result in an error which basically translates to “This content is not available on the store anymore.” 


“No, the store has not shut down (yet), you can still currently buy other games/DLCs that aren't delisted. But it's a quirk of how the Xbox 360 versions of Telltale games worked. The season pass would act as an entitlement that would let you "purchase" the DLCs for free, but the Xbox Live marketplace would still treat them as purchases. As such, like with other games, once they're delisted, you can no longer "purchase" them, but you can still re-download them if you had purchased them previously.


The Xbox One version and other newer gen versions work differently - the DLCs are linked to the disc itself, i.e, the season pass disc acts as a license for all subsequent episodes. Hence why physical copies of those versions still work today and can/will download episodes automatically. But on the Xbox 360 version, it was just an entitlement as I mentioned and not an actual license on the disc.”


Thanks to u/CorruptMemoryCard on Reddit for the explanation.
It should be noted that as of July 29, 2024, the Xbox 360 store is now closed.


TL;DR
Despite other platforms, Telltale's games on 360 don't actually entitle you to episodes directly, instead giving you a license to "purchase" the episodes for $0, hence why the episodes aren't downloadable like the other consoles.

- Yarr!

Homebrew


Modding The Console

I’d recommend buying a pre-modded unit off eBay, since the average joe would have issues performing this precise mod. If you want to do it yourself, follow a guide to install an RGH mod, and a tutorial to install all the necessary stuff. I’m not responsible if you break or brick your console from doing this. Please keep in mind that some Xbox 360 E consoles have a type of motherboard called “Winchester” (most 360 E console batches after the Xbox One’s launch) which is not moddable. It should also be mentioned that the v3-5 revisions of the Xbox 360 “Corona” motherboard (most later Slim boards) will need a postfix adapter for the NAND flashing process, to make the mod work.



Downloading/Running the Game

Using the extracted iso taken from a site (Myrient is the one I recommend and use) you can run the game via JTAG or RGH using Aurora off a usb or the internal drive. Extract the files from an iso using Xbox 360 Image Browser. You can also optionally convert the file back to iso, then to .god format. It’s up to user preference. However, converting to god is not recommended for S2. ISO is off the table since it does not work.

The Myrient dump of The Complete Adventure is reported to have a blackscreen issue. This is fixable by enabling the following in the game's settings in Aurora:
Content patching, XBLA Patching, License Patching, Fake Live.

It could also be an issue with S2, but I don’t have a unit to test it with. If it blackscreens, follow the above again. 


Getting Season Two to have all the episodes is a doozy so I’ll explain it best I can.

You need to download the iso and extract it normally. This will only come with Episode One, however, before you install the episodes, you need to update the game via Aurora. Next go to Myrient’s Digital Xbox 360 Content Archive, and download episodes 2-4. Extract the folders with a zip extractor of your choice (I personally recommend 7zip.) Using 7zip, select “extract here” and all the episodes will merge into a folder called “584114C6”, this folder needs to be placed in the games files at Content > 0000000000000000. If you’re having trouble, Modded Warfare made a great guide for installing DLC on a hacked 360.