Robert Baratheon, as a character in Game of Thrones, is quintessentially defined as someone who has let themselves go. He is not a bad person on the good/evil scale, but he's not a great ruler or example.
Take note of the following exchange between Robert and Ned:
Robert Baratheon: Lord Eddard Stark, I would name you the Hand of the King.
Eddard Stark: [kneels] I'm not worthy of the honor.
Robert Baratheon: I'm not trying to honor you. I'm trying to get you to run my kingdom while I can eat, drink and whore my way to an early grave. Damn it, Ned, stand up. You helped me win the Iron Throne, now help me keep the damned thing.
Robert was being hyperbolic here, but it is a fairly accurate summary of his activities as King of the Seven Kingdoms.
This point is reinforced over and over, it oozes out of every scene where Robert is either present or being talked about. If you look up Robert Baratheon quotes, the vast majority revolve around Robert's lifestyle.
The throne is in massive debt (to the Lannisters)
Cersei is in a loveless marriage and comments several times on how Robert sleeps around
We see Robert go out on a hunting party drunk - people's reaction to this implies that it is not unusual
Robert admits that he has changed dramatically since he took the throne:
Robert Baratheon: I swear to you, I was never so alive as when I was winning this throne, or so dead as now that I've won it.
Robert admits having grown weaker
Robert Baratheon: Gods, I was strong then.
Robert, a born warrior, is not cut out for a life of peace. He reveals as much to Barristan Selmy, when talking about the first soldier he killed and what would've happened to him had he survived the war. It's strongly implied that Robert is referring to his own life, having survived the war:
Robert Baratheon: He could have lingered on the edge of the battle with the smart boys, and today his wife would be making him miserable, his sons would be ingrates, and he'd be waking three times in the night to piss into a bowl.
Robert doesn't enjoy ruling as a King:
Robert Baratheon: The gods be damned. It was a hollow victory they gave me. A crown... it was the girl I prayed them for. Your sister, safe... and mine again, as she was meant to be. I ask you, Ned, what good is it to wear a crown? The gods mock the prayers of kings and cowherds alike.
Since neither his rule as King or his marriage to Cersei bring him happiness, Robert instead indulges in drinking, eating, whoring around and feasting - all of which leads to "such a personality (you know what I mean)".