Northbounders usually start at the end of April, beginning of May. The idea is to reach the Colorado border after June 1. There will be snow in the San Juans - but the later in June you arrive the less snow there will be. In May, unless it’s a drought year, there is usually too much to hike. You don’t want to start too late though because of fire danger in the NM forests which can close the trails. Plus it makes it more difficult to finish a thruhike.
Many northbound hikers end up flipflopping - starting in NM, then heading north in early June to avoid the usually higher snow levels in southern Colorado. (CO trails are 10-12,000’, MT is 8-10,000.) Lots of strategies for that.
Southbounders start after June 15, usually. Much earlier and the rangers at Glacier probably won’t give you a permit. You will have snow the first couple of weeks - so bring an ice axe.
Have you hiked any other long trails? I recommend that you hike the PCT before the CDT.
You can find a lot of basic cdt info at
www.spiriteaglehome.com
Ginny