Mukuro is one of those types that rub people the wrong way. He's got a fairly (if not heavily) confident attitude, although some of his haughtiness dissipates after Tsuna defeats him. Don't get me wrong, he's still arrogant, but the level lowers; one could almost say he's been humbled.
Well, for someone like him (there'll be more on that in the weaknesses section as it fits better there).
Ahem. Anyway, once you put that rather blatant characteristic aside, you're left with the remains of Mukuro. Which, honestly speaking, are probably all very bad qualities. This is something he acknowledges himself and he seems to have little care to change. He's even admitted such when, after observing his enemies, he guesses what the group's plans are. He sounds extremely certain that he's correct, yet goes on to tell his opponents that he won't reveal what they're doing. He doesn't care about his teammates or what happens to them, because he's "not that nice of a human being." This example serves as both a good demonstration of how Mukuro views himself and my next personality point: his attitude toward allies.
Going along with just the above: Mukuro's the type of person that, despite being on a "side" won't cater to it. He's got his own means and motives, so he won't go out of his way (and he probably won't even stop if you're on the way) to help someone when it doesn't directly benefit him. Even if it means having others lose respect for him, or having them hate him, he doesn't mind: it's all just a pebble on the road to his ultimate goal. And that's because, whether it's allies or enemies, Mukuro sees people as tools. They're around to get him from Point A to Point B, if he so wishes, but little more than that.
Now, I will say that's just how he sees them. There are a few of his actions that suggest he might have some deeper feelings than that (Tsuna, being the optimistic fellow that he his, confirms it), but he probably doesn't realize the full extent of it himself. So, while he sees people as tools to be used and disposed of, on a deeply subconscious level he may consider them as more. If nothing else, he has the potential to care within him, far beneath his skewed personality.
Esssseentiallllyyy... He's an ass, but not that big of one (but still is one), prefers to view himself as cruel, doesn't go out of his way for people, uses others like tools, and has the potential for compassion.
"Mukuro" is a generally nice, if timid, guy. He doesn't have a lot of confidence and doesn't like bothering people, so he becomes intimidated or uncomfortable easily. At the same time, he pretends to slowly come out of that for some reason or another. The word "shy" also comes to mind, but only in certain circumstances as he has doesn't have too much trouble approaching people.
He also has a habit of complimenting people when he (supposedly) thinks they're particularly interesting or talented. With these people he prefers to go along with their ideas and encourage them. To be honest, he can be too much of a people-pleaser (which is on purpose) and can sometimes come off as "trying too hard". Which is also on purpose.
There's... quite a few. Even if you put aside his obvious strengths (that is, his combat abilities: the Six Paths of Reincarnation) he's pretty damn cunning. Manipulations are definitely Mukuro's forte, and he can string people along through both brute force and mind games. He's not adverse to harming others, blackmailing, or just outright lying to get people to react the way he wants. In fact, Mukuro's gone as far possessing people and using their safety to get an edge, taking advantage of "fatal" diseases, beating people that are unable to retaliate, and, well, hiding his true nature. Frankly that last one... sounds nicer than it is (Mukuro's pulled off two major manipulations of this type in the series, one precanon and one in the novel. For the first he pretended to be an orphan and joined a mafia group, soon after he took possession of one of the members [who got close to him] and — using their body — killed the rest of their "Family". Mukuro then kept that person beside him until he didn't need them anymore. Now, for the second one Mukuro picks out a "just" student and pretends to be their admirer. He slowly drags the man down, changing him through carefully planned events and illusions. When the guy finally shows his true corrupted self, Mukuro reveals the plot).
Uh. Did I mention he's not a nice person yet? [cough]
At the same time, he's not wholly evil either. His methods and overall goals are evil, certainly, however he himself isn't completely gone. Mukuro's rather strange in that he's got some form of charisma. It's not something normal people can readily understand (especially if they've seen his above nature), and it's probably only something that people he's taken the time to talk to can see. In spite of how he uses others, those loyal to Mukuro? Will always come back to him. Always. Why? Because he was the one to reach out to them first. He holds out a limb for those with nowhere else to go and gives them something to believe in: him. His confidence, his words, his whole aura inspire those around him, for better or worse. To allies he's a source of relentless strength, and to enemies he's indomitable, immovable. He's a man who has seen hell six times over and it, quite simply, shows.
Which brings us back his more battle-oriented strengths: Mukuro has, literally speaking, gone to hell six times. He's died and been reborn and learned all the secrets hell had to offer. This shows in his fighting skills: The Six Paths of Reincarnation; the six realms have various abilities (illusions, possession, skill transference [he can use a possessed person's skills while in their body. But only in their body], beast summoning, and two for battle skill increases) and he's been able to master each one. I won't go into much detail since they're pretty self-explanatory, but because of his "reincarnations" Mukuro is a well-rounded fighter. He can create illusions from a distance, and fight (seemingly) expertly in close quarters. In addition, he can use his ability of possession to take over the mind and body of anyone he strikes (with a special blade).
For all of Mukuro's cunning he is far from perfect. He can be wrong, extremely so. Now, he won't admit it (more so because of a lack of care than denial probably), but there are plenty of times Mukuro's failed to correctly grasp a situation in the series. Particularly before he defeats Tsuna, where he... pretty much flings inaccuracies left and right. It's entirely possible that he's lying to throw off his opponents, however he believes in at least half of the things he says. For instance, he wholeheartedly believes he can beat Tsuna due to the has faith in his abilities. True, that in itself isn't a weakness persay, but combined with the fact that he: A.) Won't hesitate to lie about something to make it seem like he knows more than he does (to lower enemy morale); and B.) Leans more toward overconfidence than belief, makes it, uh. Pretty weak.
So, uh. The point is that Mukuro goes between telling the truth and lying, though in the latter case he doesn't care if he's correct (and usually, as stated above, is just playing a mindgame). Even for the former he can be wrong, and while that bounces off him the fact that he doesn't always see through everything? Remains. He's smart, yes, but he has a tendency to overestimate himself and underestimate the people around him.
With most of that out of the way I can explain a bit of how Mukuro's changed since then. Fundamentally he still spouts off lie/truth hybrids, but the degree lessens. After Tsuna defeats Mukuro he seems to have fewer mistakes in his fighting and deductions (they still exist though), something that can be attributed to the fact that he was once burned by his lack of caution. He observes situations more carefully, and, after gauging the one he's up against, either proceeds with the lying-truths to affect morale/concentration or saves the information for later use.
As far as his battle skills go? He's not invincible, and there are a few restrictions on his more "hax" abilities. Possession requires a "contract" of some kind (which is just being hit with a special weapon) as well as an able body, and it's also possible for his pseudo-possession/perception hijacking/illusions to be warded off with a strong will plus outside help. On top of that, if someone is extremely adverse to saying or doing what Mukuro wants? They probably won't betray that feeling.