After you have bought and installed Hearthfire, walk around in one of the towns (such as Whiterun). Sometimes it takes a few minutes, but eventually a courier shows up. He tells you he has been looking for you and has something he is supposed to deliver - your hands only. He says it looks like official business.

NOTE: If you do not get the letter from the courier, just go speak with the Jarl's steward directly. You can only buy a plot of land in Falkreath, The Pale, or Hjaalmarch.

NOTE: You do NOT need Dawnguard installed to start Hearthfire, nor is there a minimum level requirement. However, if you have not already gained the Jarl's favor, you will only receive the courier letter once you are level 9 or above.

You will receive a letter from a steward, such as:

Letter from the Steward of Falkreath
Dragonborn,
As a mark of the high esteem in which you are held by Jarl Dengeir of Stuhn, he has instructed me to offer you an opportunity to become a landowner in Falkreath. A choice piece of undeveloped land in the hold has just become available. Please contact me if you are interested in ourchasing it
Tekla, Steward to Jarl Dengeir of Stuhn of Falkreath

Note: The letter could also be from The Pale of from Hjaalmarch.

Read the letter. You will receive the objective: (find it in your mission log under MISC objectives)
Speak to the Steward of Falkreath about the land for sale.

Each Steward is in the hold capital.

Location of Stewards:
Falkreath Hold - Falkreath City (Tekla, Jarl's Longhouse)
The Pale Hold - Dawnstar
Hjaalmarch Hold - Morthal (Aslfur)

Requirements for obtaining land in each of the holds
Falkreath: Obtain the Jarl's trust by doing the "Rare Gifts" favor and "Kill the Bandits Leader" favor
The Pale: Obtain the Jarl's trust by doing the "Waking Nightmare" quest and "Kill the Giant" favor
Hjaalmarch: Obtain the Jarl's trust by completing the "Laid to Rest" quest.

If one of the stewards does not sell you land, try talking to another one.

Fast travel to Falkreath (or to the other location) and speak with the Steward. Tekla is found in the Jarl's Longhouse.

The steward should have the following dialog option:

I got your letter about the land for sale. Is it still available?

The steward will tell you something like it is a fine property, well watered plentiful stands of timber, beautiful view. Say you will take it. It costs 5000 gold to buy the land.

The Steward gives you the title to your steading. You receive the following item which is added to your inventory:
Lakeview Manor Charter

House Names
Lakeview Manor (Falkreath)
Windstad Manor (Hjaalmarch)
Heljarchen Hall (The Pale)

You then get the following objective:
Visit your new property in Falkreath Hold

And your map is updated with the location of your new land.

You also unlock the following achievement:

Landowner
Buy a plot of land