In your case, you need to use the metrics v2 API or the older one - timeseries API (less flexible). For the new metrics v2 API, the documentation is located here.
The client time response metric key for a service is called builtin:service.response.client.
An example curl call for fetching metrics for service with entity ID SERVICE-006A927F462E473C would be:
curl -XGET -H "Authorization: api-token $APITOKEN" "$TENANT/api/v2/metrics/series/builtin:service.response.client?scope=entity%28SERVICE-006A927F462E473C%29"
You can specify entities in more ways such as tags, names or management zones.
Additionally you might want to specify aggregation, dimensions, resolution, time range (from/to) or paging.
Documentation is pretty good, additional blog post explaining the API with more examples is here.