BuildSpec.yml is for AWS CodeBuild.
Go through serverless.yml and replace role names, S3 bucket names, hosted zone ids (route 53) or simply remove that section.
Checkout the details along with sample request/responses on my blog post here: https://cloudncode.blog/2017/08/08/tidbit-api-gateway-as-a-s3-proxy-cloudformation-script-with-serverless-framework/