An Azure VM is no different than any VM (or physical machine) you might create. It is completely dependent upon how much you're willing to pay for. You can refer to the current pricing here. If you scroll down to the VM descriptions you can select the # of cores and RAM you need based upon your requirements. What you didn't provide to us but you need to consider is how much storage do you need, how IO bound your process is, how fast is the actual processor and how many cores you really need. If you are IO bound then you need a later generation to get access to the SSD to keep up with the processor. If you don't need to store a lot of data then you might get away with a VM without too much storage. It is completely dependent upon what your process is doing.
But there are other issues you need to address first. It doesn't matter how many cores your machine has if your process is single threaded. The more cores you use the faster things get but not every process can use multiple cores. You should first rearchitect your app to support multiple threads if it is possible to do so. You can post questions in the forums to get help with this but ultimately it depends upon the work involved and how much "sharing" of data is needed. In a perfect world you'll scatter your work to all available processors and so the more cores you throw at the problem the faster it gets. If you cannot break your problem up into multiple threads then an Azure VM is really not going to help you at all. In fact your current machine isn't being used fully so there is no "upgrade" that would make sense. As a simple rule of thumb, if your existing machine isn't 80+% CPU utilized then getting "more" cores isn't going to help anyway.
The other thing to consider about a VM is that you're paying for it whether you're using it or not (unless you use a shared VM which has other issues you need to consider). If you simply don't want to block your main machine while your process runs then you could always get a cheap machine online somewhere and use it as a dedicated device instead. It would certainly be cheaper long term. But, again, depends upon the problems you're trying to solve.