I am not sure.. but have you looked to Section 2.2 in Applied Numerical Linear Algebra by Demmel? [1]
I think that you can get something like
kb >= d*ka* norm(B)/norm(A)
where A is the original matrix, B is the perturbed one, ka is the condition number of the original matrix A, kb is the condition number of B and d is a positive constant.
[1] Demmel, James W. Applied numerical linear algebra. Siam, 1997.