Solving problems with RDP to an Azure VM

Make sure its running! Also try shutting it down from the portal and restarting it

If you are connecting from the Internet make sure you have the correct service name and endpoint (for ASM) for the VM or if its an ARM VM ensure you have a PIP for the VM or a load balancer with a NAT rule pointing to 3389 for the VM

If connecting via ExpressRoute or S2S VPN ensure the connection is functioning and available

If you have the Azure agent installed in the VM run the PowerShell below to reset RDP and make sure its enabled and the firewall rule open:
$AzureVM = Get-AzureVM -ServiceName -Name
$AzureVM | Set-AzureVMAccessExtension | Update-AzureVM #Enable RDP

If you still cannot RDP check if there is a Network Security Group applied to the subnet the VM is in and if so remove the NSG if you think it is blocking the RDP access (or create a new one overriding to enable RDP)
Get-AzureNetworkSecurityGroupForSubnet -VirtualNetworkName -SubnetName
Get-AzureNetworkSecurityGroup -Name " -SubnetName

If all else fails you could try deleting the VM while keeping its storage and then recreating

Finally download the VHD and try fix on a local Hyper-V box