Is there someplace to upload my schedule H against my tax return?

No, you can't upload it, but you can enter the relevant information from a prepared Schedule H in TurboTax Online.

Certain services prepare Schedule H for clients, but they don't file it for you.  You'll need to file it with your return.  To enter your information in TurboTax Online:

1. Open TurboTax Online
2. Select  Tax Home in the left pane
3. On the  Hi, let's keep working on your taxes! screen, select the dropdown on the right side of the Other Tax Situations box, then select Start, Continue, or Pick up where you left off
4. On the  Your uncommon tax situations page, scroll down and select the dropdown to the right of Additional Tax Payments
5. Select  Start or Revisit to the right of Nanny and Household Employee Tax
6. Answer  Yes on the Did You Have a Nanny or Household Employee? screen, if appropriate 
7. Enter any relevant information on the subsequent screens