The x-api-key parameter is passed as an HTTP header parameter which is not added to the JSON body. How you pass HTTP headers depend on the HTTP client you use.
For example, if you use curl and assuming that you POST the JSON payload, a request would look something like (where you replace [api-id] with the actual id and [region] with the AWS region of your API):
$ curl -X POST -H "x-api-key: theKey" -H "Content-Type: application/json" -d '{"key":"val"}' https://[api--id].execute-api.[region].amazonaws.com