Staple Homes are new housing added in for the update.
Certain criteria needs to be completed before you can buy any of the houses.
One of the criteria is completing the free "A Friend in Need" quest that you have to buy from the Crown Shop under the "QUEST STARTERS" section. You will be awarded your first housing upon completing this quest but it does not count as one of the five for the challenge.
Read more about each specific house here:http://elderscrollsonline.wiki.fextralife.com/Staple+Homes
Houses cost different amounts. You can bypass the in game cost or having to do specific house required quests by paying in crowns but I advise against this unless you have left over crowns from an ESO Plus subscription. The two more expensive of the cheap Staple Homes cost about 2,000 crowns each which is the equivalent of $20. So you would be looking at spending over $50 if you bought all five needed with crowns.
Achievement tracker does not update at all after you buy a house but you can always check how many you got by going to COLLECTIONS > HOUSING > STAPLE HOMES. Just remember that the free Mara's Kiss Public House does not count as one of the five, so in the end you should have six Staple Homes.
You also don't need to do anything special after you buy the house, you can just open up the Crown Store and then fast travel to the next, buy and repeat.