In a computer system, executing programs are called processes. A process is an execution of a given program. So for example Microsoft Word is a program, a set of bits on a hard drive. When you double-click it, the operating system creates a process executing MW. The processes share the same code; but they have different data. Each process has its own private data space. The processes are separate entities accomplishing work.
It's the same with minds. Assuming you could in fact upload a brain to man-made hardware -- a proposition we'll accept for now, but which is highly debatable -- at the moment of creation they'd be two different people executing the same code and having identical data. A moment later they have each had different experiences and now have different data and life experiences from that moment on. But they are definitely two distinct people.
Another computing analogy is the fork() system call in any Unix-like operating system (linux, MacOS, Solaris, etc.) A fork() operation allows a running process to create a brand new process with a copy of the parent process's data. From that moment on, the child process has its own separate existence.
If you happen to run Chrome, each tab runs in a separate process. The parent process, the main browser process, spawns child processes -- windows and tabs -- as needed.
[Note, Not all browsers use a process-per-tab model]
It seems extremely clear to me that human mind-cloning would be the same. For example suppose you had a matter transporter like on Star Trek; but instead of transporting you, it copied you. So one you stays on the ship; and the other beams down to the planet to have adventures with whoever you find down there.
At the moment of cloning, one process becomes two, having the exact same data. Same memories, same states of biological processes. After that moment, the life experiences and biological states diverge. Two separate people.
To answer your specific question: The clone would be a brand new being, coming into existence with the exact same state as the original.
This isn't actually all that different, when you think of it, from the phenomenon of identical twins. An egg is fertilized and starts growing into a human being. The egg splits; and now you have two separate human beings. That's really the essence of it.
The real question is wether you can upload your mind to a computer. Is a mind something that exists independently of the medium in which it executes? Microsoft Word on a PC is pretty much the same program as Microsoft Word on a Mac. But is that true about humans? Is the you running on your wetware the same as you running on a supercomputer in a lab? Personally I do not think so, but that's beyond the scope of this question.