I think it's an interesting and valid character concept, in the right context. I agree with the several posters on this topic that a Paladin should embody the ideals of his diety. However, suppose this paladin's diety is of a hedonistic nature, like Bacchus or Dionysus for example. For that kind of diety, Falstaff would be the iconic ideal.
As to violations of the "rules" of Paladinhood, well the SRD defines these rules as:
"Code of Conduct: A paladin must be of lawful good alignment and loses all class abilities if she ever willingly commits an evil act.
Additionally, a paladin’s code requires that she respect legitimate authority, act with honor (not lying, not cheating, not using poison, and so forth), help those in need (provided they do not use the help for evil or chaotic ends), and punish those who harm or threaten innocents"
Whether drinking, whoring, gambling is evil depends on who one asks - and is better not debated on this board. However, it's not the classic D&D sack-a-town, sacrifce-the-maiden, heroic-fantasy evil.
Based on the Shakespearean character, only the lying and petty theft would qualify. And I'd argue that in the milieu of a game where so many adventures equate to "orc home invasions" that the "evil" of theft depends on the victims and the use to which the gains are put. As for the lying, I would question the intent - vicious slander or merely boasting when even you don't expect your audience to believe you, which makes it more a jest.
Also, as I remember the character from the Henriad, he respected legitimate authority (Hal, who was as legitimate as anyone could claim to be after the War of the Roses).
So, if the difference between the fictional character and the theoretical Paladin was that the latter actually stood up and fought against evil when it was called for, I'd be inclined to let the rest pass, dependent on the diety.
Speaking of which - you might want to consider one of the more hedonistic faiths for this character, to spin the dramtic conflict to "I could be a better Paladin if not for my weaknesses of the flesh, but doggone it, I'm supposed to have these weaknesses to emulate the one I serve!"
Just my 2 coppers,