I have never driven to Leh But I have planned to do once soon so did some extensive research. By the way I have been to Ladakh thrice and again for the 4th time I am going this year in July. So this would be my suggestion for route:
Day 0: Hyderabad to Nagpur (500km/ 9 Hrs)
Day 1: Nagpur to Jhansi (590Km/ 10 Hrs)
Day 2: Jhansi to Ludhiana (800Km/ 13 Hrs)
Day 3: Ludhiana to Srinagar (500km/ 13 Hrs)
Day 4: Srinagar to Kargil (200km/ 5 hrs) Enroute visit Sonmarg
Day 5: Kargil to Leh (220km/ 5 hrs) Enroute Visit Lamayuru Monastery, Alchi Monastery, Likir Monastery
Day 6: Leh to Nubra Valley via Khardung top (200Km/ 5 hrs) Enroute Visit Hemis monastery, Diskit Monastery. After reaching Nubra you can take camel Ride in the desert. Also Plan to start early from leh
Day 7: Nubra to Turtuk and Back (One way 85KM/ 2 hrs) Visit the beautiful village of Turtuk and take a trek through the village.
Day 8: Hunder to Pangong Tso (Merak) (185 KM/ 6 hrs this road is very much bad but scenic. Alternate you can take the Diskit Turtuk highway and then join the pangong lake road 250km/ 7 Hrs)
Day 9: Merak to Karzok (Tso Moriri) (200Km/ 5hrs) Avail Tsaka La road. Enroute Visit Kyagar Tso, Tso Kar
Day 10: Karzok to Jispa (300Km/ 10 Hrs)
Day 11: Jispa to Manali (150km/ 5Hrs) Enroute visit Vashisth Temple, Keylong, Rohtang Pass
Day 12: Manali to Delhi (550KM/ 12 Hrs)
Day 13: Delhi to Jhansi (480km/ 9 hrs)
Day 14: Jhansi to Nagpur (590km/ 10 hrs)
Day 15: Nagpur to Hyderabad(500km/ 9 Hrs)
Before Moving to Leh from Srinagar (Day 4) you can take one day for Srinagar Sightseeing.
Before moving to Nubra from Leh (Day 6) You can add one day for leh sightseeing (Shanti stupa, Thikse monastery, Leh Palace, Leh Bazaar -Shopping etc).
Before moving to Delhi from Manali (Day 12) You can add one day for Manali Sightseeing (Solang Valley, Hidimba Devi Temple)
You can add the days as per availability. For a first timer in Himalayas I feel June/July would be great time to drive because the roads will be dry very less chance of snowfall.
Let me know if you need any other details. Apart from the road conditions (lets leave that to experts, we have many who have driven the terrain many times), I can help you out in terms of hotel suggestions and sightseeing plans.