It is not possible to copy/paste your VPS to AWS.
AWS Lambda is a serverless infrastructure, I can say it is completely different to work with Lambda and an ordinary VPS machine.
In Lambda, you execute "jobs", I mean, you send your script to be processed and get its return, you don't know where it is executed, and doesn't matter at all in fact.
However, you must adapt your backend to work in this infra, adapt scripts do be Lambda compliant etc.
Regarding database, it depends on its size, but I can say it would be the simplest task for you to accomplish, you can export from one side and import from another.
You can have a look on AWS CLI that can be helpful in this task.
For your compute server, I'm afraid you will have a fresh install and then copy and configure your services and API's.
Good luck