First of all a clarification: although in English it is the same term, in Italian there is a clear distinction between 'Lombardo' and 'Longobardo'. The former is an inhabitant of the current region of Lombardy, where Milan is located, while the latter refers to the Germanic people of the Lombards. At the time of Gian Galeazzo, the Lombard kingdom was still well remembered, but no one identified themselves as belonging to that people any more. Therefore, no one would ever have thought of restoring the kingdom for real.

Gian Galeazzo was certainly a very powerful and ambitious man. And his willingness to expand his family's domains by any means at any opportunity was obvious. This unscrupulousness brought him many enemies and made the foundations of his dominion very weak: when he died, the Visconti dominion was fragmented, partly going into the hands of his own condottieri, partly the old families returned to the cities from which they had been driven out by Gian Galeazzo.

Did Gian Galeazzo want to make himself king? Yes and no. The great families of late medieval and Renaissance Italy had a major problem of legitimacy. Almost all of them at the beginning of their power had no legal basis for their power, only their money, their militia and public offices entrusted to relatives or trusted men. Such rule is fragile: what happens if the money runs out? Or do the men of trust change masters? Will power manage to pass from father to son, especially if the latter is very young? As a solution the Italian 'lords' could either try to legitimise themselves by referring to ancestors or illustrious pasts, by spending money in the arts or by buying a true title of nobility from the emperor.

Gian Galeazzo used all three of these systems. In particular, to return to your question, he made sure to recall the splendours of the early medieval Lombard kingdom to present himself as their ideal successor. For instance, he made Pavia, the ancient capital of the kingdom, his co-capital along with Milan, residing in both cities and enriching Pavia with sumptuous palaces and gardens. But this should not be read as an attempt to restore the kingdom proper, but as a propaganda operation. Also because it should never be forgotten that the kingdom of Italy continued to exist formally throughout the Middle Ages and the modern age. The emperors of the Holy Roman Empire were also kings of Italy. Anyone who really wanted to make himself king of Italy would have had to wage all-out war against the emperor.

So, in short: did Gian Galeazzo Visconti want to unite Northern Italy? Yes, he would have gladly got his hands on the whole of Northern Italy if he could. And revived Lombard kingdom? No, he probably did not dare to dream of it even though he drew on that historical legacy for propaganda reasons.