I think αὐτοῦ depends on ὅσον: ". . . as much of it as . . ." In other words, partitive genitive. The antecedent of αὐτοῦ is ὁ περὶ τῆς κατὰ φύσιν μεταβολῆς τῶν πολιτειῶν εἰς ἀλλήλας λόγος from the preceding sentence.
"Therefore, as much of it as we consider to be relevant to (ἀνήκειν) practical history and common understanding, [so much of it] we will attempt to go through in a summary fashion."
I would loosely and more idiomatically translate it something like this: "Therefore, to the extent we consider it relevant to practical history and common understanding, we will attempt to discuss it in a summary fashion."
The revised Loeb translates it this way: "I therefore will attempt to give a short summary of the theory, as far as I consider it to apply to the actual history of the facts and to appeal to the common intelligence of mankind."