Ronaldo scored a penalty and a free kick either side of his second strike on Friday to earn Portugal a thrilling 3-3 draw with Spain in Group B.
Kane, who is England's skipper for the World Cup, told reporters that his first priority is to help the Three Lions win.
"He's put me under a little bit of pressure for sure," the 24-year-old said. "He's a fantastic player and had a fantastic game but as always for me it's about concentrating on myself and the team.
"Hopefully I can score a hat-trick tomorrow and we'll be level, but it's not something I'll think about much until hopefully towards the end of the tournament.
"Maybe I've been scoring a few more goals in training. I am proud to be England captain and can't wait to get going. The team are all very close. We've done a lot of stuff for the fans so they can see what we've been up to and that's not always been the case. It's about doing it on the pitch first and foremost."
Belgium take on Panama in the other Group G tie on Monday.