How to make the keydown function work with the program in the background?

You have to register a global key in Windows. I've got a answer in the SO with what you should do:

Import the Win32 functions:

    // DLL libraries used to manage hotkeys
    [DllImport("user32.dll")] 
    public static extern bool RegisterHotKey(IntPtr hWnd, int id, int fsModifiers, int vlc);
    [DllImport("user32.dll")]
    public static extern bool UnregisterHotKey(IntPtr hWnd, int id);

Determine the ID of the key you want to manipulate:

    const int MYACTION_HOTKEY_ID = 1;

Register the key already in the main form constructor:

    // Modifier keys codes: Alt = 1, Ctrl = 2, Shift = 4, Win = 8
    // Compute the addition of each combination of the keys you want to be pressed
    // ALT+CTRL = 1 + 2 = 3 , CTRL+SHIFT = 2 + 4 = 6...
    RegisterHotKey(this.Handle, MYACTION_HOTKEY_ID, 6, (int) Keys.F12);

Create the method that will receive the Windows message to choose what to do:

    protected override void WndProc(ref Message m) {
        if (m.Msg == 0x0312 && m.WParam.ToInt32() == MYACTION_HOTKEY_ID) {
            // My hotkey has been typed

            // Do what you want here
            // ...
        }
        base.WndProc(ref m);
    }