How to change the background of a slide?

There are many ways:

1. To just change to a different plain color: Insert Square Shape > Stretch to fit slide > Style Menu (formerly Style Inspector, now Brush icon) > Arrange > Send to Back > Styles Menu > Style > Style Options > Select Color.

2. To place an image is basically the same actions but the image is selected from the Media Browser.

3. To use the background of a different Keynote template: Create/open a Keynote Presentation that uses the slide background you want > tap the slide in the Slide Navigator > Copy > Close the presentation and open the one where you wish to use the background and tap in the Slide Navigator > Paste then edit

or

Open a presentation with the background you want, duplicate a slide in the navigator, clear any items not wanted and tap Play to bring the slide to full screen > take a screen-shot (press Home Button with the Sleep/Wake Button until you hear the camera sound) > close the presentation > open your "working" presentation and insert the screen shot from the Media Browser > Camera Roll > position the image over your slide > Arrange > Move to Back.

Good luck.

One at a time - each slide you add to a presentation (using the "+" Button in the Navigator) is either the background of the template or white so you wll need to modify each as to meet your needs.

Unlike Keynote for OSX, which is more robust and allows you to assign different template backgrounds to individual slides quite easily, the iOS Keynote version is somewhat limited in both methods and choices.