I'd like to know how to replicate the process of building Ubuntu. How do I tag such a question?
I think this depends on what you mean by "building."
Currently you've tagged the question compiling. If you actually want to build your system from source packages (i.e., compiling everything from source rather than installing it from downloaded binary packages), you should keep this tag, and also clarify explicitly in your question that this is what you want. But if you're going to build your system from binary packages, you should remove compiling.
Since you're interested in creating bootable media, as bodhi.zazen has suggested it looks like you want to customize a live CD/DVD/USB from scratch. You might use the customization tag, though that tag does suffer from being quite general.
If you're attempting to create a new, Ubuntu-based distro--or you're trying to figure out how to build Ubuntu from scratch with that purpose in mind--then you could use the custom-distributions tag. Going by some of the questions with this tag, arguably it may also be used in a more general way for any situation where someone creates their own custom version of the Ubuntu live or server/alternate installation ISOs.
The live-environment tag applies if it's the desktop image you're customizing (rather than the server image, mini image, or--for older releases that had it--the alternate image). Sometimes people use live-cd or live-usb. Though live-environment is more general (which is sometimes considered bad), in this case live-environment is often more appropriate. Presumably you want your customized image to work on both optical media and USB pendrives! I wouldn't use all three of these tags, mostly because a question can only have five tags and it would be most useful for them to have more distinct meanings, but also because live-environment adequately covers the meaning of all of them.
Since you are making an iso, you could use that tag, though that is also quite general. I'd probably use it, unless I found four other tags that were all more helpful than it.
We have tags like uck and remastersys referring to specific tools that are sometimes used to create custom installation media. However, you likely won't be using either of those tools, especially since you seem to want to create your ISO from scratch rather than starting with an existing ISO. (Even if you weren't, these days I don't think those tools are as widely used as in the past.)
On the other hand, there is currently some ambiguity in your question, due to the phrase "taking the stock Ubuntu as a starting point." If, rather than starting from scratch but replicating the stock Ubuntu, you currently wish simply to unpack is contents and then put them back together with no or minimal changes, then much of what you're doing would be to extract an recreate the squashfs filesystem. In that case you could use the squashfs tag.