What's the best way to detect when the app is entering the background for my view?

You can have any class interested in when the app goes into the background receive notifications. This is a good alternative to coupling these classes with the AppDelegate.

When initializing said classes:

[[NSNotificationCenter defaultCenter] addObserver:self selector:@selector(appWillResignActive:) name:UIApplicationWillResignActiveNotification object:nil];
[[NSNotificationCenter defaultCenter] addObserver:self selector:@selector(appWillTerminate:) name:UIApplicationWillTerminateNotification object:nil];

Responding to the notifications

-(void)appWillResignActive:(NSNotification*)note
{

}
-(void)appWillTerminate:(NSNotification*)note
{
    [[NSNotificationCenter defaultCenter] removeObserver:self name:UIApplicationWillResignActiveNotification object:nil];
    [[NSNotificationCenter defaultCenter] removeObserver:self name:UIApplicationWillTerminateNotification object:nil];

}

In Swift 4.0

override func viewDidLoad() {
    super.viewDidLoad()

    let app = UIApplication.shared

    //Register for the applicationWillResignActive anywhere in your app.
    NotificationCenter.default.addObserver(self, selector: #selector(ViewController.applicationWillResignActive(notification:)), name: NSNotification.Name.UIApplicationWillResignActive, object: app)
}

@objc func applicationWillResignActive(notification: NSNotification) {

}

On your applications AppDelegate the (void)applicationDidEnterBackground:(UIApplication *)application method will be called by iOS. You can stop your timer in there.