There are three breeds of domestic cats being worked on right
now that use exotic cats to get the look of the wild.  They are
Bengals, Savannahs, and Chausies.  Other breeds exist (Safari, etc)
but they are not actively working on domesticity that I am aware of.
I'll go into each breed in detail in a moment, but here's some general
advice on hybrid breeds.
    Your most important decision will be which breeder you use.
Unlike wolf-hybrids, the 'percentage' of the cats is unimportant.  All
three breeds are working towards full domestic status (or have already
achieved it).  The goal of these breeds are to have a fully domestic
cat with the look of a wild cat -- it's not to have a high percentage
status symbol.  Breeders should be part of a breed club, and should
have their cats registered in TICA or ACFA (the reputable registries
that accept hybrids).  Breeders should breed for temperament, health,
and then for pattern/colour/size.  For more info on selecting a
responsible breeder, check out my site dealing with it at
http://nik.ascendancy.net/orchid
    So, on to the breeds themselves.
    Bengals are the oldest and most accepted of the breeds.  I own
Bengals, so I might be a little biased.  :)  Bengals are recognised as
fully domestic cats, and have Championship status in TICA and ACFA.
Bengals descend from the Asian Leopard Cat, and they are most advanced
when it comes to wild looks on sweet domestic temperaments.  Bengals
are active.  Really, really active.  It is best to have two Bengals
(or a Bengal and another active breed), as Bengals get bored and
lonely easily.  Bengals are smart.  Really, really smart.  My boys can
get through baby locks, can open the fridge and freezer, are working
on learning how to open doors, and team up to solve problems.  Bengals
can be destructive.  Not intentionally, but when they are chasing each
other around the house, they can accidentally knock over things if
they are left out.  We've lost ~10 wineglasses to the Bengal 500.
Bengals are affection-needy.  If you want a pretty sofa ornament,
these are not the cats for you.  They are outgoing, friendly,
demanding cats.  They train easily (clicker training being the only
method), and ours walk on leash and harness.  The Bengal international
breed club can be found at www.bengalcat.com, and any breeder you
consider buying from should be a member.
    Savannahs are not as advanced as Bengals.  They are only into
their 5th generation away from the Serval, and there aren't many of
those fifth generations.  Savannahs are very close to obtaining
Championship status in TICA.  They currently show in the Evaluation
Class.  Since what is currently holding them back from Championship
status is low show numbers (judges haven't seen enough of them), you
will most likely be strongly encouraged to show a pet Savannah.
Savannahs are also extremely active, and they are fairly large so
there are few places that they can't get to.  They love to jump, and
can reach some amazing heights.  They are also attention-needy, and
they do best with a companion as active as themselves.  The Savannah
breed club can be found at www.simba.cc and any breeder you consider
buying from should be a member.
    Chausies are not as advanced as Bengals.  They are into just
their 4th generation away from the Jungle Cat, and there aren't many
of those fourth generations.  Chausies currently show in the
Evaluation Class in TICA. Chausies are also extremely active.  They
are also attention-needy, and they do best with a companion as active
as themselves.  The Chausie breed club can be found at
http://chausies.org/ and any breeder you consider buying from should
be a member.

    You'll notice the common thread of these breeds is activity
and affection/attention needy.  All these breeds have been and are
being bred for exceptional temperaments, and the injection of wild
blood gives them an intelligence and curiosity not often seen in
'regular' cats.  They are not for everyone, and you should talk to as
many breeders and owners as possible before buying one.  Feel free to
email me if you have any other questions,

Orchid

Orchid's Kitties: http://nik.ascendancy.net/bengalpage
Orchid's Guide: http://nik.ascendancy.net/orchid

How about Ragdolls, Main Coons, or Norweigan Forest Cats?  Ooh...what's
that Russian one called?

All large cats are illegal to have as pets in my state, but hybrids

Oh, I couldn't understand this...do you mean wild cats??

Dee