An entry is required
here, lest the Acolytes
of the Church of the Batman
appear and perform the mysterious Bat-Spanq
upon us all.
James Gordon was originally a Lieutenant, and eventually, the Police Commissioner of Gotham City. As such, he initially found himself in conflict with Batman as he would with any vigilante. However, as time went on, he began to appreciate the presence of the Caped Crusader, and began to provide assistance to him. At first, this was merely in the form of notification via the "Bat-Signal" searchlight when The Dark Knight's services were required. Later, however, his stance eased further, and eventually he provided wholesale intelligence (and, when necessary, delayed or misdirected official police presence and inquiry) in order to preserve and protect his city's unique protector.
Eventually, his daughter Barbara Gordon was to take up the cowl as Batgirl.
One excellent picture that sums up Jim Gordon's attitude towards the Batman can be found in the graphic novel Batman: The Dark Knight Returns by Frank Miller (of Sin City fame).
Let us not even mention or picture the hideous defacing of the legend of the Dark Knight perpetrated by American television in the 1960s and 1970s. While entertaining, it should be viewed as an entirely separate story. We can be grateful that it supported Bob Kane financially, at least. Well, I hope it did.
Batman, when done right, is not funny. While there may be funny moments, the story and character are fundamentally dark and violent, a picture of tightly-controlled (and occasionally loosed) rage.