I think I see your problem. First, you need to remember that Form1 is the class name of your form and does not refer to the specific instance that is running. You would need to pass a reference to the current instance of Form1 into the Class1 instance in order for that instance to have access to the UI elements on the form. Second, you cannot access the UI elements from another thread directly.
To solve these issues, I would first ask if this is really necessary. I'm assuming this is sample code for proof of concept. If not, lose the thread and lose the class instance as well. At least don't expect to access the UI from the class instance. Instead, have the class encapsulate the work it needs to do and then have it either return the value you need from a method or have it update properties in the class that you can then access later in your methods behind Form1.
If you must do threading and you must access Form1 from the instance of Class1, you will need to pass your class instance into the Class1 constructor. Also, you will need to modify your threading call using code like what is found in this link: