Why would King Uzziah think he could do the priest's job?
The immediate answer from the text is pride: "when he was strong, he grew proud". So why would pride lead Uzziah to think he needs no priestly intermediary?
There is precedent for this of course, although Saul's sin was less brash:
8 He waited seven days, the time appointed by Samuel. But Samuel did not come to Gilgal, and the people were scattering from him. 9 So Saul said, “Bring the burnt offering here to me, and the peace offerings.” And he offered the burnt offering. 10 As soon as he had finished offering the burnt offering, behold, Samuel came. And Saul went out to meet him and greet him. 11 Samuel said, “What have you done?” And Saul said, “When I saw that the people were scattering from me, and that you did not come within the days appointed, and that the Philistines had mustered at Michmash, 12 I said, ‘Now the Philistines will come down against me at Gilgal, and I have not sought the favor of the Lord.’ So I forced myself, and offered the burnt offering.” 13 And Samuel said to Saul, “You have done foolishly. You have not kept the command of the Lord your God, with which he commanded you. For then the Lord would have established your kingdom over Israel forever. 14 But now your kingdom shall not continue. The Lord has sought out a man after his own heart, and the Lord has commanded him to be prince over his people, because you have not kept what the Lord commanded you.” ESV
Uzziah probably thought of himself more as a "David" than a "Saul", because of his success against the Philistines, and wide-ranging civil and military success:
The Ammonites paid tribute to Uzziah, and his fame spread even to the border of Egypt, for he became very strong. 9 Moreover, Uzziah built towers in Jerusalem at the Corner Gate and at the Valley Gate and at the Angle, and fortified them. 10 And he built towers in the wilderness and cut out many cisterns, for he had large herds, both in the Shephelah and in the plain, and he had farmers and vinedressers in the hills and in the fertile lands, for he loved the soil. 11 Moreover, Uzziah had an army of soldiers, fit for war, in divisions according to the numbers in the muster made by Jeiel the secretary and Maaseiah the officer, under the direction of Hananiah, one of the king's commanders. 12 The whole number of the heads of fathers' houses of mighty men of valor was 2,600. 13 Under their command was an army of 307,500, who could make war with mighty power, to help the king against the enemy. ESV
As David seemed somewhat immune from the usual rules separating priestly roles, Uzziah's pride would have made him think he had earned the same privilege and had nothing to fear.