(This guide is now fundamentally out of date, and the author, AKA me, hasn't played CKII in some time. Anyone with an up to date strategy is free to add it... that goes without saying on a wiki but applies even more so now. Of course I might reinstall CKII and add it myself.) Napoleon Complex
So you've decided to play Harold Godwinson in 1066, hoping no doubt to defeat William the Conqueror and rewrite history. There are few tasks you could have taken on in the early game which are more difficult. Harold Godwinson starts off with an impressive army of 10,000 men. However, in order to triumph, he has to defeat not only Harald Hardrada, who usually starts off with an army not far off that, but must then turn around and defeat William the Conquerors army of 12,000 veteran mercenaries.
The clincher in this situation is that Harold can actually beat both of these armies in battle; WIlliam can be beaten by holding him off at Middlesex (the river and better commanders ensures Harold'll win with around 5000 men left) and Hardradas smaller army can usually be overwhelmed by attacking him at the offset.
This is one of the many strategies which will get you killed and your throne stolen. So don't get into any fights yet. Speaking of which...
Strategies Which Will Get you Killed
1: To start with the obvious, the quickest way to die is to try to repeat Stamford Bridge by attacking Hardrada in York. This will get a large chunk of your army killed on difficult ground (you have to cross a river and fight some formidable commanders) and will leave you in no shape to fight William the Bastard. If you've chosen to attack here, you have already lost. Restart.
2: Conversely, standing your ground at Middlesex. While this will result in less of a numbers disparity and let you white peace out William after he runs away, you won't have the numbers left to defeat Hardrada, who will hunt you across England and who will eventually catch you.
3: Attacking Normandy. This sounds brilliant on the surface of things; you don't need to attack that big nasty army if you're rampaging around its homelands on the continent! Sadly, no, this will not defeat William, as he will take enough of your lands to keep in the game and to make matters worse, will start calling in French allies to take you down. Even if they don't do much directly, you can't afford more enemies, and while you and William have been swapping land, Hardrada has been gleefully lapping up fresh conquests... and you'll still be too damaged by attrition to take him on. This is a cunning strategy (as is attacking Norway) but no cigar I'm afraid.
So, with what doesn't work out the way, you're left with one option...
Don't Let William Get Ashore in the First Place
Originally, this guide recommended getting William to fight Hardrada; while still possibly a valid strategy, the more aggressive AI behaviour means that this strategy is no longer recommended. The best way to do it as of the latest patch is to wait for William to get ashore... as he lacks the transports to get his whole army across.
William will land with around half of his full army. Attack this immediately and try to position your army so that you aren't attacking across a river to do it (this may be tricky as Middlesex is surrounded by rivers and William will generally land at Middlesex). You should be able to break the back of Williams first army before the second half arrives. Wipe out both stacks, focussing on the one with the shattered morale first; you don't want the survivors joining up with the rest of Williams army (this may be unavoidable if they managed to join the tailend of the battle, but at least then you've shattered the full army). Once William's army is destroyed, you can demand he surrender.
At this point, check your forces. Do you still have around the same number of forces as Hardrada? If so, it is acceptable to save your game here, because the next part is dumb luck. You now need to confront Hardrada. But you must NOT do it directly. Hardrada is a very good commander and your forces, now even in numbers to him or even inferior, will be beaten if you attack. Instead, you must first wait for him to squander some forces in foolish siege assaults (a weak point in the otherwise good AI) and then put him in a position where he must attack (moving up to besiege the occupied provinces is a good way to do this). Try to force a battle quickly as otherwise he will bring in reinforcements and you are in deep trouble; if necessary, find a way to attack him on the plains. If things don't go your way in the battle, reload (there really is an element of luck involved at this point).
Once you've beaten Hardrada, evaluate your forces. If you've got over 3000 men left, go for the victory! (Move on to "Getting the Compensation You Deserve" section) If you have less than 3000 but still have more than 2000, you might want to strive for a victory, but be prepared for a very bitter struggle as Hardrada simply won't give up, and consider a white peace.
If you've got less than 2000 men left, forget it. White Peace out. Hardrada can easily manage to gather 1000 men before your little 2000 man stack can take anything important, and he'll wear your forces down to the point where it's just two old men slapping each other ineffectually until Svend finally wakes up to how weak his challengers are and takes it all. You might, however, have allies youcan call in to rescue the situation.
Getting the Compensation You Deserve(and Getting Those Claims Revoked)
If you have enough men, it's time to take the fight to Hardrada. The first thing you need to do is reoccupy the places Hardrada has occupied. Be wary of Hardrada landing forces during this time and hobble him if he looks like taking anything. Make steady progress; you need to preserve your army at this point as Hardrada is quite capable of raising large armies to challenge you. Once Hardrada is kicked from your shores, knock out any lingering armies and then prepare to take the fight to Norway itself.
You can attack Norway anywhere; I usually attack the south as it is closest to the English shore, and particularly Middlesex,allowing me to return in a hurry. Steadily besiege and take provinces off of Harald, taking breaks every so often to destroy Norwegian stacks and break enemy sieges (if anything gets over 1000 men, or 1/3 of your existing army, whichever applies, destroy it immediately). You'll have to break 100% warscore to finally get the hard ruler of Norway to see reason and give up his claim.
With that last victory, it is over. Harold has survived the nightmare that was the 1066 invasion of England. You've saved England from the depredations of the conqueror, broken the viking legacy once and for all, and saved both England and France from years of animosity towards one another. And possibly saved a few messy wars with the Scots as well.
Contingencies and Concerns.
You will probably get excommunicated, and suffer plots. While annoying, this is not the end of the world. You can power through it. Once you get enough money, just get a declaration of repentance sent off and get the papal censure lifted.
William may call in allies. This can seem alarming, but they seldom send large contingents and may simply feed your warscore. Keep calm and beat back any additional interlopers. More alarming is the possibility of stronger allies like the HRE getting involved. You can nip this in the bud by forming an alliance with the HRE through your heir marrying his sister. If you're lucky it might even give you a few claims on France later in the game!
If you are put in desperate straits, you can try assassinating one or both of your rival claimants. This kills their claim on your land stone dead and white peaces you out of the fighting. However, given the expense this may not be an option.
You may want to choose "Become Exalted Amongst Men" as Harolds dream. This is due to the fact that, after beating both William the Conqueror and Harald Hardrada, you will have easily breached 1000 prestige. Why shouldn't there be two English kings with the cognomon "the Great" after all?
England is a powerful nation, and Harold enjoys loyalty from his vassals. You may want to try to grab a chunk of the Baltic states before Denmark, the HRE, Sweden, Poland and Norway all take their slice; with enough determination Harold and his son can grab a good chunk of the Baltic states and even Finland before it is all swallowed up.
The tricky bit, however, is managing your sons and daughters. Marry them off carefully; if you didn't take the alliance with the HRE then choose your heirs wife carefully for maximum future political advantage. (recommendations here are welcome). Everyone else, marry them off to someone that gets you prestige, but which doesn't put them in a position where their offspring will come after your throne later on; marry them off to a 4th daughter of the king of Leon or something. And try to get Matrilineal marriages for the girls, again in such a way that they won't come hunting for you later. You do not want to have to fight off another invasion because you won the war but screwed up at the negotiating table!
Aside from the Baltic states, you can start claiming chunks off of Scotland. Keep a reserve of money back in case a crusade is called as well; England is quite capable of being the biggest contributor to a crusade if they're willing to spend enough money. Reform your economy first and expand where possible, but always keep an eye on your stability and don't anger your vassals too much or you WILL suffer for it. Other targets of expansion can include Spain, but that's a busy neighbourhood. You might be able to snatch Mallorca and Menorca though, or even parts of Sicily.
If you do pursue a colonial empire, ensure that whatever vassal gets the piece of foreign dirt has lands in England too. Otherwise they are more or less guaranteed to rebel due to the distance penalty.