In general you'll need to configure a relay agent in your router to forward/relay DHCP queries to both your primary DHCP server and the backup DHCP server. I think there is usually no way to assign one DHCP server the role as the primary and the other as only backup, DHCP requests will usually get send to both and the quickest to respond wins.
You will need to either have both DHCP servers coordinate and replicate assigned / available leases or configure them with different, non-overlapping ip-address ranges.
But that is only the theory..
AFAIK you can't use DHCP in Azure at all.
Neither the protocol, nor the Windows server role is available.
https://docs.microsoft.com/en-us/azure/virtual-network/virtual-networks-faq:
What protocols can I use within VNets?
You can use TCP, UDP, and ICMP TCP/IP protocols within VNets. Unicast is supported within VNets, with the exception of Dynamic Host Configuration Protocol (DHCP) via Unicast (source port UDP/68 / destination port UDP/67).
https://support.microsoft.com/en-us/help/2721672/microsoft-server-software-support-for-microsoft-azure-virtual-machines?wa=wsignin1.0
The following roles are not supported on Microsoft Azure virtual machines:
- Dynamic Host Configuration Protocol Server
- Hyper-V (Hyper-V role is supported in Azure Ev3, and Dv3 series VMs only)
- Rights Management Services
- Windows Deployment Services