API Gateway Security Using AWS Services like ACM

Need to restrict my API gateway hits based one AWS services like ACM
I need suggestion or idea how to achieve?.
Only valid client only permitted to hit API gateway

My proposed solution is below
Prerequisite is need to create certificate using ACM - public and private key
1. Client will invokee API using AWS API Gateway
2. While invkoing API gateway from client it should send private key along with payload
3. When receiving the request to API gateway, it will validate the request using public key
4. If valid request, AWS API Gatway forward the request to Our Backend Service

Provide best solution