Is it permissible to borrow money from one's father, with his knowledge that the reason for borrowing is to rid oneself of ill-gotten gains, and based on his trust in the son's ability to repay from money previously acquired through illicit means?

You must disclose your true financial situation to your father so you do not deceive him. If someone acquires unlawful wealth and then repents, they must give it away in charity. However, if they are in need, or to pay off debts, or to start a permissible business, it is permissible for them to take from it what suffices their need, and then donate the remainder. Based on this, you do not need to borrow from your father. Rather, it is permissible for you to take from this wealth what is sufficient for you to start a business. It is better to consider it a debt upon you that you will give away in charity whenever Allah grants you affluence.