September or October can be preferred as weather would be clear and scenic colours there are mesmerizing. Even roads are mostly feasible for bike ride. I did it in September 2018. All the best.
Either June or October first week. If you want to see ice wall in the high mountain passes then make a plan in June but in that time the condition of road won't be very good, you have to cross a lots of water crossing. On the other hand, if you are more interested about the blue sky and good paved road then better to plan at October first.
September is one of the best time to go to Leh as this time offers some of the best sights of Ladakh and better road conditions (than amid the Monsoons). As the peak season (June-August) is over, now the rates of the hotel are decreased.
The Ladakh temperature during this time is Maximum 21° C, Minimum 5° C. The climate is cool and wonderful chill begins to look in. It in some cases you can find snows amid late September. Get a chance to enjoy the crystal clear water with glimmering sunbeams falling over the surface and making it sparkle.
Also at this time, the road conditions are much better than the usual. At times there can be snowfall which can cause a roadblock.
Visit some of the best parts of this region like the mesmerizing Pangong Lake and enchanting Tso Moriri and get a chance to learn about the culture and the beauty of this region. Also visit the famous monasteries and the stupas here while you pass through the highest motorable road.