You're probably aware that the CT is really high in elevation (average over 10000), and whenever that happens, you need to be extremely cautious for the first couple of days when planning your hike with regard to having adequate water and not trying to go quickly. Additionally, the odds are that snow could stick around until early July in some places, and the weather in CO is always unpredictable.
I've only done a couple of small sections of the CT, but my experience has been:
Close to Denver: dealing with foothills and pretty dry, but busy. I'd try to steer clear of the first couple of sections.
West of Leadville: Pretty good water sources, really high, but spectacular views (some of the best hiking I've ever done). Camping and wildlife seem to be the best/most abundant in this area.
Closer to Durango: Decent hiking, but you might could do better.
If I was going to plan two weeks for myself, I'd probably try to start near Breckenridge/Copper Mountain and head west. I can't really break it down into daily mileages, but starting at section 8 (http://www.coloradotrail.org/segments.html) I'd probably try to get through about section 14. Ideally, you could arrange a drop-off and pick-up from family or friends, but you could do well with an out & back as well.
And in general, and I speak from a great deal of personal backpacking experience, you'll enjoy the trip a lot more if you have someone with you.