Kyrgios, 19, edged French 13th seed Richard Gasquet 3-6 6-7 (4-7) 6-4 7-5 10-8 after saving nine match points at the All England Club on Thursday.
Gasquet tipped Kyrgios for grand slam success and a top-five ranking, but the big-hitting world number 144 is aiming higher.
"My goal is to become the number one player in the world," Kyrgios said.
In an entertaining second-round clash, Kyrgios hit 86 winners to go with 44 unforced errors as he battled past Gasquet.
He labelled some of his performance 'unbelievable' and was pleased to have fought to stay in the contest.
"I played some unbelievable tennis today. He was coming up with some really good shots as well," Kyrgios said.
"There's plenty of opportunities he could have taken. I came out on top, I'm really happy."
A meeting with another young star in Czech left-hander Jiri Vesely awaits for Kyrgios.
Both are junior grand slam champions, with 20-year-old Vesely winning the Australian Open in 2011 and Kyrgios victorious at Melbourne Park two years later.
Vesely stunned French 24th seed Gael Monfils in the second round and Kyrgios knows he will be facing a huge challenge.
"He had a massive win today, winning in five sets," Kyrgios said.
"He's also young. He's on the rise. He's got a really aggressive game, big serve. Big hitter as well."