Non locking UI code in WP8
Minor


You should get rid of the Class suffix. There is no value in attaching the type as a suffix to the name. 
You don't need to write this everywhere to access members.
You have an Elapsed property which is not used.


Bugs/Major


Your Reset() method will break the stopwatch because you create a new timer but you do not re-attach the timer tick event handler.
You have a Reset() method which sets up the timer but you also do this in the constructor which is some code duplication and in fact has introduced a bug (see previous point). DRY - Don't Repeat Yourself. Make use of a common method which can get called in both places.
Instead of "polling" your stopwatch in a continuous loop your StopwatchClass should expose an event like TimeUpdated to which any interested party can hook up to.

In general this can pose problems when interacting with UI because if the callback happens on a different thread then you need to marshal UI access across to the UI thread. However you have used a DispatcherTimer which is evaluated at the top of every dispatcher loop - which means the callbacks are happening on the UI thread and it is safe to access UI elements from within the callback (see the MSDN example).

So basically you can expose an event which gets fired on the timer callback and whatever code hooks up to it can update the UI. 
In general MVVM has established itself as a very useful pattern. Which means that rather than updating UI elements directly you would update an observable view model to which the UI is bound. This decouples the UI better from the data and decoupling is always good.
The timer is guaranteed to not fire before the interval has elapsed - but there is no guarantee about how long after the interval has elapsed it will fire. This means your stopwatch will start falling behind the real time as you assume each tick is 1 second while in fact it might be 1.01 sec or so. This means after 100 seconds you could be easily 1 sec off or more. A thread time slice in windows is typically around 10ms so I'd suspect the timer to be around that accuracy. Although on Windows Phone you might get better performance (couldn't find any documentation in that regards).

The way to overcome this is to take a timestamp at Start and compute the difference to the current time on each tick. While this will probably also give you an approx. 10ms accuracy it will be constant while your solution will accumulate an error on each tick.

This will also alleviate the need to compute the seconds, minutes and hours since passing as you can get that from the Timespan.


With the changes from above your code could look like this:

public class StopwatchClass : IDisposable
{
    DispatcherTimer _timer;

    public int Interval { get; set; }

    public int Seconds { get; set; }
    public int Minutes { get; set; }
    public int Hours { get; set; }

    public bool IsRunning { get { return _timer.IsEnabled; } }
    public TimeSpan Elapsed { get; set; }
    public List&lt;TimeSpan&gt; Splits { get; set; }

    private DateTime _StartTimestamp = DateTime.MaxValue;

    public event EventHandler TimeUpdated;
    private void OnTimeUpdated()
    {
        var handler = TimeUpdated;
        if (handler != null)
        {
            TimeUpdated(this, EventArgs.Empty);
        }
    }

    public StopwatchClass(int interval)
    {
        Reset();
    }

    void timer_Tick(object sender, EventArgs e)
    {
        var timeSinceStart = DateTime.Now - _StartTimestamp;
        Seconds = (int)timeSinceStart.Seconds;
        Minutes = (int).timeSinceStart.Minutes;
        Hours = (int)timeSinceStart.TotalHours;
        OnTimeUpdated();
    }

    public void Start()
    {
        _StartTimestamp = DateTime.Now;
        _timer.Start();
    }

    public void Stop()
    {
        _StartTimestamp = DateTime.MaxValue
        _timer.Stop();
    }

    public void Reset()
    {
        Seconds = 0;
        Minutes = 0;
        Hours = 0;
        Splits = new List&lt;TimeSpan&gt;();
        _timer = new DispatcherTimer();
        _timer.Interval = new TimeSpan(0, 0, interval);
        _timer.Tick += timer_Tick;
    }

    public void Split()
    {
        var timeSpan = new TimeSpan(Hours, Minutes, Seconds);
        Splits.Add(timeSpan);
    }

    public void Dispose()
    {
        _timer.Tick -= timer_Tick;
    }
}