The week right after New Year's is supposed to be one of the slowest of the entire year. You'll see a little bump around the long MLK weekend, but the marathon doesn't cause crowds to rise appreciably.
I would suggest a little research on the competitions at Wide World of Sports. January and February are prime times for bother cheerleading and dance troop competitions. Both events can cause rises in attendance, mostly on the Friday and Monday that bookend the competition weekend. You also may want to re-think a stay at an all-star resort during any competion period.
We were at WDW last January on a Sunday through Friday (afternoon). There were no events during our arrival weekend, and crowds were very low the entire time. The weekend of our departure did have an event at WWoS, and it was VERY easy to spot the large groups of girls making their way through the Magic Kingdom on Friday. Attendance was noticably up compared to the rest of the week.
Prior to our trip, I actually found a web site that listed national cheerleading and dance competitions. So I knew in advance that nothing was going on the weekend of our arrival. Don't remember the site, but an Internet search would probably turn up something. Or, a call to WWoS might provide the same info.
January is a very good month for touring the parks, as long as you are prepared for the weather. You can get some very beautiful, sunny days with 70 degree temps. But there will also be days in January where the high doesn't get out of the 50s.