Your fortress really only needs 3 things: Food, Fun, and Fermented Beverages.
Food
Dwarves need to eat to survive (duh!). If there is no supply of food in the fort, dwarves will quickly stop doing any assigned jobs as they attempt to hunt vermin to survive.
Dwarves aren't happy when they're forced to eat vermin. A farm of plump helmets is a good way to stave off hunger indefinitely, as long as you have seeds to plant. If you don't cook plump helmets, they will leave seeds when eaten, allowing infinite sustainability.
Booze
Take a gander at any one of your dwarves' profiles. See that last line? "and needs alcohol to get through the working day". It's that important. Every single one of your dwarves runs on alcohol.
No alcohol means unhappy dwarves who are forced to drink water. If they can't drink water, they'll die of thirst. You cannot rely on your stocks screen alone to determine if you have enough booze, because dwarves have one (and only one) rule of drinking -- you can't drink from the same barrel another dwarf is currently drinking from. You may have 200 units of alcohol in your fort, but if all of it is stored in a single barrel produced by your legendary brewer, you'll lose dwarf after dwarf as the line for the barrel gets longer and longer.
So there's two things to keep in mind for sustaining alcohol levels. First, production, which is easy - simply brew the plump helmets you've been growing to eat -- they are a very versatile mushroom. You'll even get the seeds back.
The other thing you need is enough barrels to store the booze. This usually means wooden barrels from a carpenter, but you can make metal barrels as well at a forge. And, recently, stone pots, from a craftsdwarf workshop fulfill the same function. If you don't have excess stone for as many pots as you need, you're a) not digging down far enough, or b) stuck above an aquifer.
Happiness
If the excrement starts to hit the rotary device and dwarves start dying off, your ability to recover is tied directly to your dwarves' collective happiness. If too many dwarves become severly unhappy, you risk a tantrum spiral that culminates in the death of your fort. I'll simply quote from df.magmawiki 's page on Tantrums:
Dwarf A, a craftsdwarf, gets the urge to build a mysterious
construction, but unfortunately there are no 'body parts' or shells to
be had.
A goes berserk, kills dwarf B, and wounds dwarf C.
A is then struck down by dwarf H, making an unhappy thought for A's friend dwarf
E.
B's loved one dwarf D, and friends E and F, get a strong unhappy
thought.
D, going to help C, throws a tantrum.
He kills C in his rage, giving another unhappy thought to C's friend F.
F, now horribly
unhappy from the death of two of his friends, throws a tantrum
destroying E and D's beds.
E, now very unhappy from two of her friends
dying as well as losing her bed, goes melancholy and commits suicide
by jumping down the well, giving an unhappy thought for D and J; and
contaminating the only water source.
D, despite the happy thought for
fighting (and killing) C, is overwhelmed by unhappy thoughts and
tantrums again.
This time he destroys a bridge, drowning F in the moat
(as it has no ramps), and then punches dwarf G.
F's friends, I and J,
both get an unhappy thought.
E's body rots, causing J to tantrum. He
punches G, hospitalizing him.
D once again punches someone, this time
I - cutting his lip.
Without fresh water, G dies a slow death of
dehydration.
I, even more angry due to being punched, punches D back.
D, finally being overwhelmed, goes stark raving mad.
I, angry about
being D's punching bag, punches J, and J punches H out of anger.
I, Unable to wash his lip with clean water combined with all of the
rotting corpses, succumbs to infection.
D dives into magma, creating a
unhappy thought for D's friend, H.
H, in the wake of death and Miasma
finally goes berserk and finishes the fortress off.
The more happy thoughts dwarves have, the less they care when their lover / child / bff dies a horrible, horrible death.
Luckily, Dwarves are simple creatures, and it is relatively easy to keep them entertained. The wiki even has a page dedicated to this.
A quick suggestion would be to flood your dining hall with statuary and decorations, give each dwarf a bed of their own, and don't let them go hungry or thirsty.
Best Answer
It's not going to be easy.
As you may know, Dwarf Fortress has no cheating system built in other than what can be done through modding/hacking.
You are probably going to need DFHack. DFHack edits Dwarf Fortresses memory directly and should be able to remove the offending Dwarf. The specific dfhack command to kill a unit is "exterminate". Used with no argument, this command will list all possible targets to kill. An alternative for the sadistic is to use the DFHack command "liquids-here", which will drop 7/7 magma on the location highlighted by your cursor.
Disclaimer: As far as I know, using DFHack is not considered illegal as it has been discussed and distributed on the bay12forums with no negative response from the author.