Use the following, assuming that C:\process.exe is a console application:[1]
PowerShell -NoP -C "C:\Process.exe; exit $LASTEXITCODE"
Note: If the executable path contains spaces, enclose it in '...'; if it contains ' itself, either escape the enclosed ' chars. as '', or enclose the path in \"...\" (sic) instead - see this answer for more information.
In order to get a process' exit code, you must wait for it to terminate.
Start-Process does not wait for termination by default, unless you add the -Wait switch - but then you'd also need -PassThru in order to return a process-information object whose .ExitCode property you'd need to report via exit.
Direct execution of the target executable, as shown above, (a) results in synchronous execution of console applications[1] to be begin with and (b) automatically reflects the process exit code in the automatic $LASTEXITCODE variable variable.
Without the exit $LASTEXITCODE statement, the process' exit code would be mapped to an abstracted exit code: 0 (success) would be reported as-is, but any nonzero exit code would be mapped to 1 - see this post for more information.
Either way, PowerShell's (powershell.exe's) exit code will be reflected in cmd.exe's dynamic %ErrorLevel% variable.
[1] If your application happens to be a GUI-subsystem application, you'll indeed need a Start-Process-based solution; as you've noted in a comment yourself, the solution would be:
PowerShell -NoP -C "exit (Start-Process 'C:\Process.exe' -Wait -PassThru).ExitCode"