IMPORTANT

This tutorial only applies to the default R6 1.0 body parts.
Other body parts (2.0, ROBLOX Girl, all R15 body parts, etc.) are retrieved from Roblox's asset delivery pipeline and cannot be modified. This will also apply to EVERY instance of the body parts you replace, not just your own.

This tutorial does not cover UV mapping for Roblox's avatar texture format.
View this DevForum post to learn more.

Requirements

What you'll need:

A desktop that can run Roblox Studio
3D modeling software that can export .obj or .fbx files (Blender)
The BTRoblox browser extension (Chrome, Firefox)

Getting the mesh

Open Roblox Studio and create a new file (or open an existing one). Insert a MeshPart and click on the folder icon next to "MeshId" to import your .obj/.fbx file as a mesh.

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Once the file has been uploaded, a string of text "rbxassetid://[numbers]" should appear. Those numbers are the asset ID for your mesh. Copy the numbers, open your browser, and enter the following URL into the address bar:

https://create.roblox.com/store/asset/[PASTE YOUR NUMBERS HERE]

That will take you to the Creator Store page for your mesh. Now, using the BTRoblox browser extension, click the download button at the top right and select "Download as .mesh".

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Replacing body parts

If you're using the default Roblox client, you'll have to repeat this step every time it updates.
Third-party launchers such as Bloxstrap can save mods without having to add them back after every update.

Once you've downloaded your .mesh file, locate your Roblox folders in a file explorer. Examples below:

Default client: C:\Users\dynai\AppData\Local\Roblox\Versions
(select the most recently created version)
Bloxstrap: C:\Users\dynai\AppData\Local\Bloxstrap\Modifications

Body parts: After locating these folders, go to the ...\content\avatar\meshes folder within your Roblox files. Bring your .mesh file to this folder and replace the body part of your choosing.

Heads: Repeat above, but instead go to ...\content\avatar\heads. Only the default 1.0 head can be replaced (head.mesh).

Tip: add an extra character to the names of old body parts instead of deleting them, just in case you need to restore them!

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And now you're all set! Enjoy your weirdly disconfigured body wherever you go!
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(It looks really weird here because I didn't set it up for classic clothing. Again, view this DevForum post to learn more.)