The only mistake is the mixture of formal and informal register.
The request 'May I speak to Alice?' and the response 'Speaking' are formal English, while the greeting 'What's up?' is very informal. You would never use these phrases in the same context.
Here's an example of a formal phone conversation, for example, in a business context:
Hello. May I speak to Alice?
Speaking. Who's calling, please?
This is Dan Wilson from CLA Plastics.
Hello, Dan. How can I help you?
And here's an example of an informal conversation between friends:
Hi. Is Alice there?
This is Alice! Who's that?
Hey, Dan. What's up?
I hope that helps.