See the attached chart. It partly depends what you consider to be cold.

http://www.incatrailperu.com/inca_trail_weather.html

The generally accepted "best time" to go is May-Sept mainly because it is drier rather than because of the temperature. We did the Lares trek in August at probably the coldest time of the year but it was dry with sunshine the whole time. As long as you are prepared with the right clothing temperature will not be an issue. However, I doubt I would have enjoyed it as much in the rain and mud.

Hello,
I trekked in early September and it did rain one afternoon evening, more like extreme, pouring rain, and it happened to be right after we got to Dead Woman's Pass, so it was all downhill stone steps for a couple of hours in pouring rain, I was worried about slipping on the smooth steps. That was bad, but mud would present its own uncomfort as well. The rest of the time was pleasant, could be cool and was hot and humid with the gnats, I would advise dressing in layers to cope with either extreme if you are in the May or September range.

Yes the best time is from May to October. To check for availability on permits go to the official link below.
http://www.machupicchu.gob.pe/

I went in June and you do need to book as early as possible during the high season as permits are limited. The days were sunny with nice temps and the nights were cool but quite bearable.

I went in October 3 years ago. Only had rain for one day, but there was a lot of fog covering up much of the landscape on the last day of hiking through the rain forest. If I was to do it again I would pick May, because the rain season is coming to an end and everything is still lush and green during that time.

The best thing you can do on the hike is to pack as light as possible, preferably just a light camera and your water bottle

I just got back, we went Apr 16-19. It rained on and off the second and third day, no downpours though. It made for some hairy descents but it wasn't that bad, it just made it more interesting. The coldest it got was at Dead Woman's Pass when it got windy and the second night, but not cold enough to break out the thermals. Sweats, socks and a knit cap kept me warm at night. Days were comfortable, most of us just had T-shirts on and a light jacket when needed.
I agree with the last post, late April-May would be the best time. Dry season also means not as green. Getting a little wet is a slight inconvenience, the vistas were oh so worth it. The cheap ponchos they sell may look like a joke but buy it, they will keep you and your day pack dry.