If I start a process on the console, and then ssh in with the same credentials is there any way I can view the process?

You question is a little vague. What information do you want from the process? If you just want things like memory usage or processor usage, you can use "ps" or "top" after you ssh back in. If you want control over the process as if you had executed it from the command line after you ssh back in, you can use something like "screen".

Which process & where you're doing ssh... please specify.
Meanwhile, a process can be identified by it's process ID i.e. PID. And you can check processes using ps command. To check processes of a user username, invoke:-

If what you are asking for is a way to see what that process is writing to the console where it was started, then the answer is "no" unless the terminal emulator for that console provides some hook to let you see what is displayed. If what you want is a way to look into the internal workings of that process, then yes, you can tell gdb or strace to attach to that running process and both view and control it, but gdb in particular isn't likely to be of much use unless you've got the source code in hand and the program was compiled with debugging symbols included.

Thanks for the input.

As rknichols assumed I was searching for a way to view the process, not information about it.

I run a lot of iterative computations that take hours or days to complete. I'm looking for a way to check in on them.

You might pipe the output through tee and write a copy to a file you could examine. Just watch out for output buffering. By default, output going to a terminal is line-buffered (flushed after each newline), but sending the output through a pipe would make the default block-buffered (typically 4KB blocks), so the output would be delayed until a block is filled. The program doing the computation would need a way to select how output is buffered, or you might take a look at the unbuffer utility (part of the expect package).

screen

That looks like it should work well. I'll give it a shot.