Start the test reporter. Make sure that you put these lines at top of your spec_helper.rb.
Manually tirgger fist test report (your project token can be found at Settings > Test Coverage)
It should show following message:
Now you should be able to see the report showing in Code Climate. Sometimes it can take a few minutes.
3. Modify run_tests and make it report after each deployment
When you run rspec without CODECLIMATE_REPO_TOKEN, even you have test reporter started, it won't send test report to Code Climate. In other words, you can control when to send report by giving token or not.
Modify the rspec command in lib/capistrano/tasks/run_tests.cap
I assume you always run complete test suit with this technique.
Now, test will be run before every deployment and test report will be sent after that.
Note: you can put your token in run_tests.cap for simplicity. But I rarely commit any key or token into repostory. Instead, I put it in a git-ignored, separated config file. Than read and assign the value in config/deploy.rb.