Can I further ferment store-bought kim-chi?

First, as noted in Sneffel's answer, "real" kimchi is eaten at a variety of fermentation levels.

It is possible the fermentation would resume in store-bought kimchi, but only if it is raw and unpasteurized. A lot of store-bought kimchi is pasteurized (depending on where you buy it), which would kill off the bacteria that would typically be used for fermentation. Kimchi that is not pasteurized would typically be sold in a refrigerated section and would perhaps have a container/lid that would allow for gases to be released as fermentation continues. (The only real way to tell would be labeling or contacting a manufacturer.)

In general, I would not advise trying to resume fermentation on a store-bought product, as you don't really know the conditions it was made under or whether continued fermentation would proceed safely. (Chances are that commercial kimchi would be processed in a way to ensure its safety, even if stored at low but not refrigerated temperatures, but again, this is difficult to know.)

Kimchi isn't that hard to make to begin with, so if you can't find what you like in the store, I'd suggest just making it yourself and letting it ferment to the stage you prefer.

You don't need to bury it. Just leave it, sealed, on your counter for a few days.

A couple of notes: the kimchi you are getting is "real kimchi". Kimchi is eaten at a variety of maturity levels, anywhere from a few hours to a month or more. (A year would be entirely too long.) Some Korean families would make their own kimchi; others would buy it. And it wouldn't ordinarily be buried in a clay pot. In a glass jar in the cupboard would be more usual.

Not sour enough for you? Add in a Tbsp of sugar, tip and shake to dissolve. Then give it 3 days or so for beasties to convert to CO2 and acid. check daily for off-gassing then tamp down again to keep everything submerged.