The Cech Republic international joined The Gunners from London rivals Chelsea last June, and was left underwhelmed by their lack of success in what looked like a promising campaign midway through.
A 2-0 win over Bournemouth in December saw Arsene Wenger's side go top of the Premier League log, only to slip up and allow Leicester City to clinch their maiden top flight title.
However, Cech is determined to sustain a title bid in the new campaign in order tp put his barren first season behind him.
"We were disappointed," he told Sky Sports. "We thought that we were good enough to challenge for the title and to be in the title race until the last game.
"Unfortunately, Leicester did remarkably well and it was a historical achievement for them.
"We couldn't match them but we finished second, which was the best finish for Arsenal for a long time.
"There were a lot of positives from our last campaign but obviously we would like to carry that on and improve for the next season because we finished empty-handed and that is not what we wanted.
"We started well with the Community Shield but obviously you want to add some important trophies to that.
"The aim for the next season is to add some silverware to the cabinet and we will definitely work for that."
The 34-year-old shotstopper is gearing up for what he believes will be an exciting 2016/17 season, with Liverpool their first task of the campaign, while he tips the big six to all challenge for league honours.
"It's great to really have a test in the first game and then you can kick on your season if you start it the right way," he added.
"Liverpool will be one of those teams who will want to challenge for the title, having played well with Jurgen Klopp after his arrival.
"We expect a big challenge because [Manchester] United have been improving their squad, they've changed their manager and they'll be hungry for success.
"Manchester City with their new manager will do the same thing and let's not forget Liverpool and Spurs and us, I think it will be a very exciting season and will be very challenging for many reasons.
"Chelsea will definitely be there with Antonio Conte... We're in for an exciting season."