You should reach delhi first and then start your journey to auli. Best is to get board on telangana express (12723). It will take you around 24 hours. From delhi to reach Auli is to take a bus from Delhi to Rishikesh. This is an overnight journey.
From Rishikesh, either you can take a private cab direct to Auli (9 hours) or a shared taxi till Joshimath (8 hours). A more fun way would be to take a shared cab till Joshimath and from there, get on the Second Longest Cable Car in India from Joshimath to Auli. This cable car is 16 kms long, takes 22 minutes and blesses you with breathtaking views of the Garhwal Ranges of Himalayas and then take a cab from Rishikesh to Joshimath. Do remember that cab/bus operators do not drive after sunset on the roads between Rishikesh and Joshimath, they might promise you but it is risky.