It's fine, but omit the second comma.
Where you have consecutive adjectives like this, the comma is used in place of and. You could write first generation and working class engineering student, but you wouldn’t normally write first generation and working class and engineering student.
A comma after both first generation and working class would suggest that they and engineering were all modifying student equally. But what you probably want to convey is that you are an engineering student who is both first generation and working class.