Speaking to reporters at a news conference ahead of this weekend's Hungarian Grand Prix where he seeks a record-extending ninth victory, Hamilton, 39, said: "It's game on.
"We're fighting for wins and we're chasing -- and we're going to try and win as many races as George (Russell) has mentioned or to compete for as many wins as possible.
"It might turn out this weekend that the Red Bull is still light years ahead or that McLaren is still there this weekend. Who knows?
"But I think what we've shown over the last few races is that with determination, with real focus and by persistently chasing for perfection, and improvement, you can make a difference.
"And, so, we are united as a team and we're going to try and make sure that we can end the season on a high."
Hamilton added: "And if we can lift the team up in the ladder of constructors... if we can progress – and we're not too far off the drivers up ahead of us -- then that would be great."
Fresh from his emotional and long-awaited record ninth victory at the British Grand Prix, and the record 104th of his career, Hamilton was in upbeat mood at the Hungaroring, one of his favourite circuits, where he has won eight times and claimed his most recent pole position last year.
Russell, 26, who won in Austria a week before Hamilton's Silverstone triumph, hopes to see the team extend their winning streak by completing a hat-trick on Sunday – and then going on to continue winning races for the rest of the year.
"Our goal for the rest of the year is to win as many more races as we can – and if we can have good weekends here and in Spa (Belgium) next weekend, then I'm sure we can be fast everywhere," said Russell.
"We can win more races."
