You start in the south-west corner of the map, and need to lay siege to The Snake in the north-east corner. In general, you should build your economy, especially by creating a trade post and harvesting as much iron as possible. Once you can muster a large enough army and catapults, you can lay siege to the enemy. Your opponent doesn't pose a threat himself, instead relying on the reinforcements of The Pig to assist.
First, build your castle wall to protect against attacks. The signpost is to the south east, where The Pig sends hostile archers and macemen from time to time. Once your defense is up and at least two walls thick, they should pose no threat.
Your main source of income should be building iron mines, and selling surplus iron at the trade post.
Your food stocks are initially stable, but you will want to have additional dairy farms and hunters posts (in order to get additional food sources). If left as-is, you will also get a surplus of flour, requiring you to build additional bakers.
Aside from the archers you have on the perimeter, you need to build up a force of at least 30 archers, macemen and crossbow infantry (preferably each). The army can be funded by selling surplus iron and other supplies. You will alos need to build an engineer's guild, and recruit 10 engineers to build at least 5 catapults (7 may be more ideal). You can optionally recruit enginerrs to build a portable shield to protect the catapults.
When you feel for force is large enough, advance your catapults into firing rage, and have them knock down the walls - aim for the defense towers with the siege engines first.
When the towers are down, you can advance your army. Note that the Snake has an array of Killing Pits at the base of his mountain, and a second array between the walls.
Once you breach both walls, charge the castle with all your forces.