How to Remove Dog Impurity: Islamic Guidelines
Question

How is purification from the Najasah (impurity) of a dog done? Is it essential to wash seven times or is once enough?
Praise be to Allah.Steps for Purifying Vessels Licked by Dogs
Muslim (279) narrated that Abu Hurayrah (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “The purification of the vessel of one of you, if a dog licks it, is to wash it seven times, the first time with soil.”
Muslim (280) narrated that `Abdullah ibn Al-Mughaffal (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “If a dog licks the vessel of one of you, let him wash it seven times and rub it with soil the eighth time.”
In these two Hadiths, the Prophet (peace and blessings of Allah be upon him) explained how purification from the impurity of a dog is to be done , which is by washing the vessel seven times, one of which should be with soil. Both are required.
Ibn Qudamah (may Allah have mercy on him) said in Al-Mughni (1/73):
There is no difference of scholarly opinion concerning the fact that the  impurity of a dog must be washed seven times, one of which should be with soil. This is the view of Ash-Shafi`i. (End quote)
An-Nawawi (may Allah have mercy on him) said in Al-Majmu` (2/598):
The scholars differed concerning the licking of a dog. Our view is that it makes whatever it licked impure, and the vessel must be washed seven times, one of which must be with soil. This is the view of the majority of scholars. Ibn Al-Mundhir narrated the view that it is obligatory to wash it seven times from Abu Hurayrah, Ibn `Abbas, `Urwah ibn Al-Zubayr, Tawus, `Amr ibn Dinar, Malik, Al-Awza`i, Ahmad, Is-haq, Abu `Ubayd and Abu Thawr. Ibn Al-Mundhir said: This is also my opinion. (End quote)
Shaykh Ibn `Uthaymin (may Allah have mercy on him) said:
If the impurity is somewhere other than on the ground and it is the  impurity of a dog, then it must be purified by washing it seven times, one of which must be with soil. (End quote  from Majmu` Fatawa Ibn `Uthaymin, 11/245)
Understanding the Role of Soil in Removing Dog Impurity
It is better if the first washing is the one that is done with soil, but if it is done otherwise that still achieves the purpose and purifies the place.
An-Nawawi (may Allah have mercy on him) said in Al-Majmu` (2/598):
It is recommended to use the soil in the first washing, and if that is not done then it is better to do it in any other time other than the seventh. But if it is done in the seventh it is permissible. It says in the reports in the Sahih: “seven times” and “seven times, the first time with soil”; in one report it says “the last time” instead of “the first time.” In other reports it says, “seven times, the seventh with soil” and “seven times and the eighth time sprinkle it with soil.” Al-Bayhaqi and others narrated all these reports, which indicates that it is not essential that it be done the first time, rather that it be done one of those times. (End quote)
Different Methods for Using Soil to remove Dog Impurity
Washing with soil may be done in several ways:

Washing with water then sprinkling soil on it
Sprinkling soil on it and following it with water
Mixing soil with water and washing the vessel with it.

This was stated by Shaykh Ibn `Uthaymin in Sharh Bulugh Al-Maram, Hadith no. 14.
Additional details can be found in the answers below:
How to Remove Dog’s Impurity from Carpets
Is Touching a Dog Prohibited?
Does Touching a Dog Break Wudu?
And Allah knows best.