Usually sending keystrokes to another application is a relatively straightforward operation. You find the process, get the main window handle and then send the keys. For chrome, this is a little more tricky, as activating the main window doesn't get you in the correct state to send the keys. After doing some digging, I found this question on stackoverflow. Basically, you need to find the Chrome_WidgetWin_1 window and take the window immediately following that window. Activate that window and then you can send the keys to properly open a new window in Chrome. The code for this is:

private class NativeMethods
    public static extern bool SetForegroundWindow(IntPtr hWnd);

    [DllImport("user32.dll", SetLastError = true)]
    public static extern IntPtr FindWindow(string lpClassName, string lpWindowName);

    [DllImport("user32.dll", SetLastError = true)]
    public static extern IntPtr GetWindow(IntPtr hWnd, GetWindow_Cmd uCmd);

    public enum GetWindow_Cmd : uint
        GW_HWNDFIRST = 0,
        GW_HWNDLAST = 1,
        GW_HWNDNEXT = 2,
        GW_HWNDPREV = 3,
        GW_OWNER = 4,
        GW_CHILD = 5,

private void button_Click(object sender, RoutedEventArgs e)
    IntPtr chromeWindow = NativeMethods.FindWindow("Chrome_WidgetWin_1", null);
    IntPtr chrome = NativeMethods.GetWindow(chromeWindow, NativeMethods.GetWindow_Cmd.GW_HWNDNEXT);