Applying for a new doctoral program after expulsion
I don't think your goal is impossible, but it will be difficult to achieve. I assume that your past infraction was serious or you wouldn't be in the state you are in. You can do one thing now, and you must do another thing later, however. 

Now, you can find ways to behave in a way that shows you learned from your experience, have paid for your sins, recognize them for what they are, ameliorate the bad effects they caused, and that it can never happen again. This is harder (even) than it sounds, as some acts might not be repeated, but an underlying attitude might lead a person to do other things, not exactly the same, but also quite bad. Narcissism, for example, can often lead to a lot of bad behavior of various kinds. Also, thing things you do should leave some visible record that clearly shows a change. Even better if other people can attest to your good character now and in the future. 

The thing you must do later is to be completely honest with any institution that you apply to. It won't be enough for you to promise to behave better. But the things you have done in the interim should speak for your change without your comment.

Even then, of course, there are many who won't believe in your change. You need to make it real change, both for your future chances and for yourself generally. 

Good luck. 



Note that I mention narcissism as an example only. Lack of empathy is another. But I'm not trying to attribute either to you.