Normally performance testing of mobile application is not required as mobile applications have only one user and if you're happy with application load time, responsiveness, rendering times, etc. - you should be good to go.
If your application is being updated frequently and it is relatively big and complex you might want to automate the steps and set some thresholds which define maximum operation completion time. In that case check out the following tools:
If your application explicitly relies on backend server, i.e. it's intensively using web or database to display data you may want to check the performance of the backend i.e. when several hundreds or thousands users are using the application in the same moment. In that case you can simulate the traffic which is being sent from multiple mobile devices, collect and analyse backend metrics and key performance indicators. You can do it with i.e. Apache JMeter - see Load Testing Mobile Apps. But Made Easy for details.
All mentioned tools are free and open source.