The 3 plots of land available to purchase are found in
- Falkreath
- Dawnstar
- Morthal
Each plot of land will cost you
5000 gold and will require you to perform some degree of miscellaneous quest for the Jarl of that location.
Falkreath requires you to kill some bandits and the leader, after which you can purchase the land from the Jarl.
Morthal has you solving the
Laid to Rest quest. Which involves a fire at Hroggar's house.
Dawnstar requires you to complete
Waking Nightmare quest which involves the citizens experiencing terrible nightmares.
Through this quest you can earn the
Skull of Corruption, a Daedric Artifact (which is missable if done incorrectly.)
See the Oblivion Walker guide so you don't end the quest wrong and miss out on a trophy.
Depending on the current standing of the civil war and whether you joined the StormCloaks or Imperial Army, you may have to go kill a giant as a quest before purchasing the plot of land as well.
After purchasing the land, you will be given a miscellaneous quest to visit the plot. From their you can begin construction of your home.