Cook led his team to draws in the last two test series against West Indies and New Zealand and will be working alongside a new coach Trevor Bayliss in the five Ashes tests starting on July 8.
"From where we've been if we did win the Ashes that would be remarkable, something I'd be very proud of. It would be my best achievement," Cook told reporters.
England left on Saturday for a four-day training camp in Spain where they will meet Australian Bayliss for the first time.
"We're in a unique situation that 11 days before an Ashes series none of us had met the new coach before, so going away seemed to be the best option," Cook said.
England made a terrible start to the summer when they slumped to 30 for four on the first morning of the opening test against New Zealand at Lord's before an aggressive partnership of 161 between Joe Root and Ben Stokes led their recovery.
"To me the summer started with that partnership," said Cook who became England's highest test run-scorer during the New Zealand series.
"The way they played was very un-English -- a counter-attack scoring at six runs an over rather than trying to scrape through -- and from that moment on there was a sea change in everyone's mentality."
England won the Lord's test and lost the second match at Headingley but the team's resurgence continued in a thrilling one-day series against the Black Caps which the hosts won 3-2 playing an exciting style of cricket under captain Eoin Morgan.
"English cricket feels good at the moment," Cook said.
"Being away from the international game for a couple of weeks while the one-dayers have been on, the way Eoin led that team, the cricket they have played has been fantastic to watch.
"I couldn't have been more proud. Five or six of those guys who were heavily involved in that team are going to be heavily involved in the Ashes and absolutely we can use some of that."
Australia won the last Ashes series 5-0 on home soil in 2013-14 but have not lifted the precious urn in England since 2001.
(Reporting by Ed Osmond; editing by Amlan Chakraborty)
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