Pius IV inherited a council that had already run, in fits and starts, for nearly two decades by the time of his election in 1559. Trent had opened under Paul III in 1545, been suspended more than once amid war and plague, and by the late 1550s its eventual completion was no longer something the church could take for granted. Pius IV reconvened it for its third and final period in 1562, and it was under his authority that the council finished its work and closed in December 1563.
He confirmed the council's decrees the following year, in 1564, giving them formal papal authority rather than leaving them as the acts of an assembly. That single administrative act is why Trent's decisions became binding Catholic doctrine and discipline rather than remaining a set of proposals, and it is the reason his pontificate is remembered less for any personal theological program than for successfully closing a process three previous popes had failed to finish.
Much of the practical administration of turning Trent's decrees into functioning church policy fell to his nephew, Charles Borromeo, whom he had created cardinal and who is generally credited with the energy behind the reforms that followed, while Pius IV himself supplied the papal authority that made those reforms binding.