Well, you actually could mine coins on a VPS some nine years ago, but it’s not viable any more, especially for Bitcoin. Here are a few reasons why:
1. Bitcoin mining is a parallel computing task, thus you need a GPU or even better a specifically designed piece of hardware called ASIC. Even though you can rent GPUs in the cloud, do the math before diving into this venture.2. Bitcoin mining difficulty has dramatically risen over the past years, rendering CPU mining impossible, check this graph out: https://blockchain.info/charts/difficulty?timespan=all3. Most providers will suspend your account for mining, as your server will consume 100% of CPU share 24/74. Most providers specifically disallow mining on their equipment
If you still want to mine coins with a VPS and found a provider that allows it, take a look at altcoins, especially those that can be mined with CPU.