Caching layer in front of AWS Elastic Container Service
Solution 1:
You don't put Redis in front of ECS but you can make your ECS task consult the Redis cache to see if the required result is there. If yes return it to the caller, if not compute it, return to the caller and also store it to Redis for the next time. That's a pretty standard pattern but requires support from the Application running on your ECS - it needs to know how to talk to Redis. Or to Memcache (simpler and may be enough for simple caching).
Alternatively you can put your REST API behind a CloudFront CDN and enable caching on that level. If done right CloudFront will recognise similar requests and return the cached responses right away without even reaching out to ECS. This can be transparent and doesn't need the App support.
Hope that helps :)