Read the intro and the segment on "Yoritomo and Tokimasa, the Hōjō, and becoming shogun" in this older answer I wrote (or just read the entire thing: a couple paragraphs in the 2nd half are still useful for context).

To add the situational specifics:

Why a Fujiwara? Sanetomo's social status rivaled that of the exalted Fujiwara ministers at court. Hence, the elites in Kamakura felt the need for someone of similar—or even better: higher—status to replace the prematurely deceased Sanetomo, ideally an imperial prince. But Retired Emperor Gotoba denied handing over one of his sons—he likely saw an opportunity to start a diplomatic power game to expand his influence over the shogunate—so they settled for the most rational next best option, a child of Fujiwara (Kujō) no Michiie, who held amicable relations with the shogunate, and whose chosen son incidentally also happened to be the great-grandchild of Yoritomo's sister. (However, to what extent the latter fact actually mattered is up in the air.)

If you're interested in more on the whole social divide and Tenno vs. shogun (etc.), I suggest browsing the FAQ for relevant questions.