What is the ruling on my studying at the institute when I have not fulfilled my father's condition of not using my own learning method?

There is no harm in studying at the institute as long as it does not violate the Sharia. You must fulfill your father's promise and obey him in what is good. If you believe your method is better, you should explain it to him to convince him. As long as you have not fulfilled your promise, you must seek forgiveness and repent.