Here's a good rundown on how Muellers firing could go down. Not surprisingly, it's not very straightforward...
The first issue is what supposed misconduct might constitute the basis for Mueller’s removal. The standard for firing special counsel, remember, is “misconduct, dereliction of duty, incapacity, conflict of interest, or for other good cause, including violation of Departmental policies.” It is hard to see what Mueller has done that would warrant termination under this standard. Trump might claim that Mueller has a conflict of interest because the firm he left represents the Trump family is some matters. But the Justice Department’s own ethics experts ruled that there is no conflict and “that Mr. Mueller’s participation in the matters assigned to him is appropriate.” Perhaps Trump would claim that Mueller has a conflict because Comey and he are friends and “brothers in arms,” but it is hard to see how this constitutes a conflict in the investigation of the Russia matter. Maybe Trump will come up with some other reason under the regulation.
Second, under the regulation, the decision to terminate lies not with Attorney General Jeff Sessions but with Deputy Attorney General Rod Rosenstein, who is the acting attorney general for purposes of the Russia investigation. So in the first instance, presumably, Trump could order Rosenstein to fire Mueller. Rosenstein would then have to decide whether he believed the reasons Trump gave were adequate under the regulation. If so, he could carry out the order. If not, and if he refused to do it, Trump could fire him—or he might simply resign in the face of Trump’s order (more on which below).
Third, if Rosenstein resigns, that raises a question of who becomes the acting attorney general. Succession in the Department is, to a point, outlined by statute: where the attorney general and deputy attorney general are unable, the associate attorney general “shall act” as attorney general. Otherwise the attorney general “may designate” the solicitor general and the assistant attorneys general, “in further order of succession,” to act as attorney general. That means it could go down the line until an assistant attorney general did not resign and instead carried out the President’s order. (Succession is complicated by the fact that, after Rosenstein, there are only two other confirmed officials in DOJ: Associate Attorney General Rachel Brand, and Acting Assistant Attorney General for the National Security Division Dana Boente, who was previously confirmed to be the U.S. Attorney for the Eastern District of Virginia. A tad more on this below.) If the officers after Brand also resign, then an executive order on DOJ succession recently promulgated by Trump would control. (This barely noticed executive order would potentially assume great significance if Trump fires Mueller.)
Fourth, there are a number of hard questions about whether Trump could circumvent the regulations—either ignore them or abrogate them—and fire Mueller himself. The argument at bottom is that all executive power is vested in the President; law enforcement is at the core of Executive power; there is no contrary statutory directive, as in Morrison v. Olson; and the Special Counsel rule is just a regulation promulgated by the Executive Branch, not a law, and is thus ultimately subject to change or disregard by Executive order. On the other hand, there is this important point made in United States v. Nixon concerning Special Prosecutor Jaworski and the regulation then in force, 38 Fed.Reg. 30739, which required “extraordinary improprieties” for his removal:
https://www.lawfareblog.com/if-trump-fires-mueller-or-orders-his-firing