Giovanni is a rather enigmatic figure in the series. His role is important but we are given only glimpses of his personality. Through meeting him in the game and seeing him in the anime we can deduce what he is like. Immediately we see the most prominent aspect of his personality, that being a harsh and demanding boss. He is the leader of a crime syndicate and as we see how he interacts with the members of the group we are shown that he can be ruthless and intimidating. Most members show fear when addressing him, but also respect, showing that he can operate without being completely heartless. Obviously, these individuals respect him knowing that in turn, he is willing to take care of them on some level. (Where it may be money or maybe his favor.)
There are some instances in the games where we find that Giovanni is also arrogant. In the first battle upon meeting him in the FireRed and LeafGreen versions he announces: "So! I must say, I am impressed you got here. Team Rocket captures Pokemon from around the world. They're important tools for keeping our criminal enterprise going. I am the leader, Giovanni! For your insolence, you will feel a world of pain!" Breaking down this quote where he says "-I am impressed you got here", this shows that he is prideful in his power and control over the organization. Thinking that, since he has so many followers and they have powerful Pokemon at their disposal, it would be almost impossible for anyone to reach him or foil his plans. We see where he says "-you will feel a world of pain!" that he is sure of his power as a trainer.
When Giovanni is defeated, however, he does not continue to pursue the fight nor does he revert to more violance as the answer. Upon his defeat as the Team Rocket boss he declares "I can't face my followers. I have betrayed their trust.-" This shows that his relationship with Team Rocket is mutual and that there is another side to his personality, that he is honourable. Though he may be the demanding, prideful boss he also knows when he is defeated and how to rightfully treat the victor. Having so much power and money could easily corrupt him, especially as the leader of a Pokemon Gym, but granting other trainers the badge that they deserve shows that he also is willing to give respect just as much as he likes to earn it from others.
His personality shows that he is more complex of a character than we see at first glance. We want to immediately see Giovanni as the evil villain but in time we see that he is simply doing "business" and is able to share deep bonds with other characters.