One important thing to note is that Suslov never wanted power (he preferred the role of the man behind the scenes). In Twilight of the Red Tsar he only agrees to become leader because there aren't any other options. Also, in order to take control of the USSR instead of Brezhnev Suslov would have to be General Secretary, not Premier. The post of General Secretary was the most important job, and it's a much more logical choice for Suslov (who had been on the Secretariat since 1947). All that said, if Suslov were to take power a good POD would be that he doesn't become sick in 1963, enabling him and not Brezhnev to lead the coup against Khrushchev.
Suslov would have been a very different leader than Brezhnev. Suslov was the de facto leader of the Stalinist faction, and as such he opposed pretty much any and all attempts at reform. He was also far less tolerant of corruption than Brezhnev (he was a believer in Leninist asceticism), so anti-corruption drives similar to those launched by Andropov would probably occur earlier. Suslov's other major passion in domestic affairs was maintain collective leadership. The semi-breakdown of collective leadership that occurred during the late Brezhnev years (with Kirilenko being demoted and Podgorny being sacked) will play out differently. I suspect Suslov would still maneuver against Podgorny and the other liberals, but unlike Brehznev he probably wouldn't take the opportunity to expand his own power. Finally, there would be much harsher measures taken against dissidents than those of Brezhnev, mainly because Suslov was terrified of both Western influence and the Party losing power (again, the actions Andropov took while in power are a good facsimile of what Suslov would do).
In the foreign policy sphere Suslov had two goals: keeping the Soviet Union from going to far with détente, and maintaining Soviet power over the Communist world. Suslov was not completely opposed to détente, but believed that it should be as limited as possible, and that the Soviets should never lose sight of the fact that they were in conflict with the West (he was particularly opposed to military détente). Treaties like SALT and the Helsinki Accords are probably a no-go under Suslov. When it comes to maintaining Soviet power over the Communist world it's important to note that Suslov often favored solving crises like the Prague Spring or the Polish emergency with political rather than military solutions. There would be nothing like the Brezhnev doctrine under Suslov, and the Prague Spring might not end in military invasion (but any attempt by Dubcek to break free of Soviet control would be met with harsh resistance).
Suslov died a little under a year before Brezhnev, but the power struggle after Suslov's reign might be different. For starters, Andropov only became Second Secretary because Suslov's death opened up the position. Without this opportunity Andropov might not get on the Secretariat, and thus not be in the running (no one in the Soviet leadership would be willing to give power to the KGB Chairman, and the position of Second Secretary was an unofficial prerequisite for becoming General Secretary). Konstantin Chernenko would also never reach the top of the hierarchy, because he owed his rise to Brezhnev's patronage (in fact Brezhnev violated several unofficial rules of the Soviet leadership to bring Chernenko power). Andrei Kirilenko also owed a lot of his rise to Brezhnev, so he'll be less powerful ITTL (also his severe mental decline precludes him from leading). So who would take over? One person who could gain power was Boris Ponomarev. He was a protégé of Suslov and longtime head of the International Department of the Central Committee (which exercised control over the international Communist movement). With Suslov in power Ponomarev might be made Second Secretary in the 1960s or 1970s, with the job of maintaining and strengthening Suslov's power in the Party and Central Committee (essentially the same job that Kirilenko and Chernenko did for Brezhnev). This leaves him very well set up to succeed Suslov and continue his hardline policies.