Kane scored his 97th and 98th Premier League goals for Spurs to break Teddy Sheringham's record as Tottenham eased to victory at Wembley.
England striker Kane has been talked up as someone who could break Jimmy Greaves' all-time Tottenham record of 266 goals and Alan Shearer's all-time Premier League record of 260.
But he is not focusing on future milestones, telling BT Sport : "Four-nil in any game is a great result, Everton are a good side and we played well, we created space and took our chances.
"It's something I'm very proud of but it's on to the next one, just keep going and these boys are great at setting me up. We've got to just keep winning games as a team and hopefully I'll keep scoring.
"I might have been offside [for his first] but sometimes you get them sometimes you don't. It went my way so I'm happy."
On the possibility of one day surpassing Greaves, Kane added: "Let's see what happens. I try to keep my goals a bit shorter than that. Month-by-month I'll set a target, but who knows what will happen."
Son Heung-min was the best player on the pitch as he scored the opening goal, set up Kane's first and hit the post.
The South Korea forward has been involved in more goals at Wembley this season than any other Tottenham player, scoring seven and assisting four.
However, Son accepts trying to match Kane is a futile task.
"I think we played very well, it's not just me because the team-mates help me that's why I score a lot of goals at Wembley," he said.
"It's impossible to beat this guy because he's one of the best strikers and one of the best players but he works really hard. He's younger than me but I'm learning from him."
Spurs have a full week off before returning to Premier League action against Southampton on January 21. Kane and company are level on points with fourth-placed Liverpool, who take on runaway league leaders Manchester City on Sunday.