Grabowski graduated from Ludwik Solski Academy for the Dramatic Arts in Cracow in 1974, and shortly afterwards started to perform at Juliusz Słowacki Theatre. Two years later he starred in the play Przepraszam, czy tu biją. During his studies in Cracow, Grabowski debuted in the full-length movie, Odejścia, powroty. He played only supporting roles in films until 1989, when he played one of the main characters in the movie Kapitał, czyli jak zrobić pieniądze w Polsce.
Grabowski gained widespread popularity in 1999 when he started to play the main character Ferdynand Kiepski in the TV series Świat według Kiepskich. The show is still on air.
Grabowski's other notable appearances include the music video for "Pokaż mi niebo," a song by Polish band Feel, and the third season of Polish version of the French game show Fort Boyard. He is also known for his Polish dubbing for characters in movies such as Harry Potter, Toy Story 3 and Chicken Little.
Grabowski was awarded a Medal Zasłużony Kulturze Gloria Artis in 2007 for outstanding contribution to Polish culture.