Building view transform matrices
To solve the problem, we will flow three steps, first, calculate translation matrix, second, calculate rotation matrix, third, get transformation matrix.

Because translation is a affine transformation, we need to use a 4x4 matrix to represent translation. I find you use column vector post-multiply matrix. the result of the translation matrix is same as yours.

Translation matrix = 

Then, same as your idea, we use basis vectors(same as unit vectors of the coordinate axes) to calculate rotation matrix.


basis z vector: we can get rotated basis z vector by using a vector from VRP to point A, and normalize it.

z = b(VRP) - a = [5,-2,1] - [1,1,1] = [4,-3,0] ,then, normalize to [0.8,-0.6,0]
basis x vector: we know basis vector is a vector perpendicular to z-up plane, we get the x vector by cross product up and z

x = up x z = [1,1,1] x [0.8 ,-0.6 ,0] = [0.6, 0.8, 0]
basis y vector, y is prependicular to z-x plane.

y = z x x = [0.8 ,-0.6, 0] x [0.6, 0.8, 0] = [0, 0, 1]

finally, the rotation matrix is:

Rotation Matrix = 
   

The 4x4 matrix is:

Transformation matrix = 

The transformation matrix is a matrix from Object Space to World Space, we should invert this matrix to get World Space to Camera Space matrix.(Thanks for @JarkkoL correcting me).

Refer to 3D Math Primer for Graphics and Game Developmen.




Following your request for clarification: I just combine the translation matrix and the rotation matrix, actually, It's the result translation matrix(T) multiply rotation matrix(R). 

The rotation is a 3x3 matrix, we can extend the matrix to a 4x4 matrix with adding a dimension, the 4x4 rotation matrix is:



And the translation matrix is:



So, now we can calculate the transform matrix by translation matrix multiply(T) rotation matrix(R):

Cause the matrix multiply formula is too big for the 4x4 matrix, we just look at the 1st row, In the transform matrix(M), the M11 element equals to T11*R11+T12*R21+T13*R31+T14*R41, cause T12, T13, R41 is 0, so, the M11 equals to R11 right? And in the same way, M14 equals to T11*R14+T12*R24+T13*R34+T14*R44, only R44 equals to 1, so, the M14 equals to T14. so, I just simply put them together in my last answer. For the reason why we put the translation matrix on the left, you can check on the book that I gave you in my previous answer on page 91(translating first and then rotating).