How to clear search history AppStore

What do you mean exactly. Just tax the X in the search field. If you tap the empty search field and it still takes you back to the last search that you performed, close the App Store completely, and then come back and tap the X and that should clear the search history.

Double tap the home button and look for the App Store in the multitasking screen, then drag the app thumbnail preview up off the top of the screen to close the app.