First start the quest,you can see snow in the following 3 places:
North of Lumbridge，North of Rimmington，North of Falador

Find head now imp

chat with him, he will give you a ice amulet.

wear the necklace and right-click to ‘Operate’ it，There will be a lovely snow imp near you,，you’ll be able to click on your imp to check whether you’re getting hotter or colder – follow the imp’s directions，just believe him, till you find jack frost, and then talk with him.

Then use ice amulet to jack frost, you will see a period of flash, at the end, you will see a Santa Claus, talk with him, then you finished the quest.

At last, you may get a santa suit comprising:

• Santa costume top
• Santa costume legs
• Santa costume gloves
• Santa costume boots

From: http://www.usfine.com/Christmas-Event-chasing-jack-frost-guide-news-77361.html