How can I mine Monero or other altcoins using Azure Cloud Services? Is there any way by which I can mine Monero or other altcoins using Azure Cloud Services.
Sure - you can run cpuminer-multi on Azure or AWS, and it will CPU mine. You will need to find a pool to mine with.

Alternatively, if you want to solo mine, then you merely need to run a full Monero node, and from within bitmonerod you use the start_mining command. Details on the options you can pass to that command are available via the help command within bitmonerod.