What is the ruling on my studying at this institute, knowing that I lied to my father about the source of the money from which I paid part of the fees?

Obeying parents is obligatory in matters that do not contradict the Sharia. The questioner's action involved two prohibitions: disobedience by violating the father's prohibition against buying the bicycle, and lying, which is forbidden by Sharia. Therefore, one must repent to Allah for these two sins and seek the father's forgiveness if no greater harm will result from it. There is no objection to continuing studies at the institute.