The K-1 visa and the Green Card Lottery are very different immigration devices in the United States:
The K-1 visa is a United States non-immigrant visa for the fiancées of United States citizens to come to the US in anticipation of their marriage.
People who are married to US citizens are eligible to apply for permanent residency within the United States and after a short period of a couple of years they can apply for citizenship as well.
If the happy couple decides that they want to be married in the United States and are currently engaged they can obtain a K-1 visa until the non-US spouse can apply for a Green Card.
K-2 visas are used for the children and dependents of the immigrating spouse and must be applied for separately.
The Green Card Lottery is a yearly Green Card assignment scheme that gives out about 55,000 Green Cards all over the world.
Applications are typically accepted in October and the winners are announced in May. Anyone who can meet the two simple but strict education and location requirements can apply for this visa and include their spouses and dependents on the application as well.
The Green Card Lottery grants eligible winners with a Green Card which is an immigration status.
How are they Related?
Really, the two applications and statuses are not related at all, but that is perfectly fine. Because they do not interfere with each other you can apply for both at the same time and there will be no conflict in the USCIS.
However, if one has the chance to apply for a K-1 visa, it is advisable to pursue that path to a green card and citizenship than the Green Card Lottery.