So, I'm assuming you've heard of the Orion's Arm Universe Project if you're asking a question like this, but if you haven't:
J-brain, Jupiter Brain
at the bottom of the page, there's this paper:
[The physics of information Processing super-objects: Daily Life among the Jupiter Brains, By Anders Sandberg]
Depending on what you're looking for in this answer, this paper may or may not actually be helpful. If you're looking for the very rough underlying physics of an absurdly futuristic theoretical megascale engineering project, yes, this might be helpful.
if you're just looking for enough of a description to use as the basis for a story, then I can give an slightly educated guess:
Other than building a computer the size of a planet?
Other than the logistics of coordinating the construction of a structure millions of times larger and more complex than anything civilization has yet produced?
Other than inventing and mass producing the chips/ brain equivelents that would be capable of operating in the 300K+ gas giant environments you're describing?
Other than inventing, building, and maintaining the heat dispersal mechanisms sufficient to feed, power, and cool a "mind/brain" larger and possibly more complex than all of earth's ecologies combined?
Just living on a godlike alien mind/ transapient civilization the size of a gas giant sounds difficult enough to me.