Can I record my performances?

Yes, by using the Recording feature in Serato DJ Pro.
Click the [REC] on the top left of the Serato DJ Pro screen. The Recording panel will be displayed in the middle of the screen.
Click the [REC] button in the panel to start/stop recording.
Click the [SAVE] button when the recording stops to save the recorded file.

For details, refer to Serato DJ Pro Manual.
You can read the Serato DJ Pro Manual [serato.com].