What is the ruling on using the internet without informing one's father? And is it permissible to use it for seeking knowledge if that is unlawful?

The questioner should thank Allah that his father is keen on protecting him from the Fire, and he should apologize to him for using his money in disobedience to Allah and without his permission.

If the father’s objection to using the internet is due to its use in disobedience, then there is no harm in using it with its Sharia-compliant controls.

However, since the internet was an entry point for Satan, it is safer for the questioner’s faith to avoid it and substitute it with ready-made, Sharia-compliant, safe scientific programs, for a believer is not stung from the same hole twice.

If the father forbids the use of the house's internet except with his permission, then obedience to him is obligatory, and it is not permissible to use his money except with his good will, and obedience to parents is obligatory in matters that are not disobedience.

Seeking knowledge is not limited to the internet, and most scientific books and شرح (explanations/commentaries) are available on CDs.

However, if accessing the internet is through a personal mobile phone, then there is a detailed explanation for that.