It is necessary to have an experienced player who gives rules advice and
can do the mental arithmetic for attacks near-instantaneously - preferably
spotting likely options while people are dithering; and it has to be well
understood that this player will not lie when acting in these capacities
(which is awkward, because being an experienced player they will be lying
most of the rest of the time.)

Be devious. It's not really about the game mechanics; you won't win with a
straightforward 'what do I want, what does he want, should I spend to
block that' approach; it's about playing the other players - like
Diplomacy, but with the bonus that the joint victory rules mean that there
might not actually be a stab coming.

[I think this is one of Illuminati's strengths. Backstabbing is easier
because sometimes you _are_ telling the truth; in Diplomacy, you always
intend an eventual betrayal.]

Accumulate special cards. They can't be taken from you and they can't be
easily counted. Resist the urge to use them early; an early advantage is
useless if recognised by the other players (in all these diplomatic games,
how strong your position is is not as important as how strong it appears.
A large early advantage will probably get you beaten on mercilessly, where
in a game like Settlers it could only be advantageous (unless you play
with people who implement trade boycotts.)) 

Accumulate money. Encourage other people to spend theirs on blocking each
other's attacks and attacking each other - encourage two players to think
of a given strong group as 'theirs'; early in the game formulate a couple
of 'we all pay equal' alliances to block someone from some 'important'
group (a high income group for the Gnomes is an excellent choice. In fact,
everyone needs income, and you might pick it up particularly if you go
next turn. But things like a Wierd group for Discordia or the
Semiconscious Liberation Army for Bermuda will do) - you don't care about
the group, but it gets the idea in people's minds. Later they will
formulate such alliances, and unfortunately you will be short of money on
your Illuminati group, or just not interested - avoid pissing them off by
being generally helpful (using the IRS to tax the current public enemy is
great - kudos for not taxing the other players, kudos for taxing the
enemy, and more money.) Take moderately high income groups 'in return' for
foregoing groups that help you with your special victory conditions; "I
could go for the Mafia, but it's another Violent group, and I think you'd
oppose me; but I'd like something this turn - does anyone mind if I go for
the Yuppies?" If the other players go for it (and with enough arguing over
the Mafia, you look like you're giving in), you've got the income and
reinforced the idea of those expensive blocking alliances. Avoid the
high-profile mega-rich groups like the Oil Companies; foment expensive
squabbles over them and then suggest that Cthulhu be allowed to blow them
up 'for the common good' whilst they belong to someone unpopular (and now
Cthulhu is paying to do what you want done under the mistaken impression
that he might get 8 kills during the average game, and you get kudos for
being a reasonable diplomat!)

Encourage the other players to drink _moderately_ and form grudges. They
will be more likely to listen to your mild voice of reason telling them
why they want to do what you want.

[Booze is a little unsporting, but actively encouraging grudges is not.
A begrudged player is not thinking rationally about who has 3 special
cards and a big stack of money.]

Once you have your big stack of money, wait your chance for the vulnerable
arm of Power Structure; provided you know that the special card that makes
a roll a 12 isn't lurking, you can lop it off (and pick up the one group
from the pool that lets you win.) Easiest with Discordia; 2 Wierd groups
is nothing, someone will have an arm with 2 more in, and there's one in
the pool. Enough cash and it's yours. Alternatively, do a 'groups for
money' deal with Zurich, after reading the rules on giving away money and
groups _very very_ carefully.

[My favourite moment; near the end of a tense 6-player game, everyone
trying to work out a joint victory - at least 2 other players would have
the cash Zurich needed on their next move. People were going out for
joint victory discussions all the time, and I told Zurich
that I would have the money ready. Round comes the turn, and I
ask the question "so in return for those groups, I give you this stack of
money. Deal?" "Deal" she says, and we trade. "That's a joint victory,
then?" "Well, maybe. You might want to count that first. It might be a
little short."

The trouble with that trick is you can pull it only once with any given
set of players.]

Basically, sit tight; let the other players fight; stockpile money; never
look aggressive until you give the one almighty punch that wins.