A basilisk head is a very rare drop from Basilisks, which requires 40 Slayer to kill.
Before it can be mounted in a Skill Hall within a player-owned house, it must be stuffed by the Taxidermist in Canifis, where she will charge 4,000 coins and return a stuffed basilisk head to the player. They can then mount the head using 2 teak planks. When players take it to the Taxidermist, she will tell the player "My, he's a scary-looking fellow, isn't it? He'll look good on your wall!"
The head can also be sacrificed to the Dark Altar north of Arceuus, granting 2,500 Prayer experience. This does not require any favour.
For an exhaustive list of all known sources for this item, see here