How do I open these doors in the room with the dead king?

Solution 1:

The weapon depicted on the door is the key to open it.

For example, this door...

...is opened by the Thistle sword.

You have to purchase (or acquire) that weapon, then activate the door (the item does not have to be equipped, just in your inventory). You then receive the weapon inside.

Solution 2:

Each is unlocked by a weapon. All these weapons can be bought in the store. Some examples:

• Gelder (dual) unlocks nehte (sword)
• Infinity blade (sword) unlocks aegis armour
• Ricochet (dual) unlocks eavet (dual)
• Halfstar (heavy) unlocks Aran (helmet)
• Imperial (heavy) unlocks kreiger (shield)

You don't need to equip the weapon.