Implement this key cipher

Python 3, 130 bytes

Thanks to @Rod for pointing out a bug

from random import*
def f(x):l=10**len(x);k=str(randint(0,l-1)+l)[1:];print(''.join(chr(ord(i)+int(j))for i,j in zip(x,k))+'\n'+k)

A function that takes input via argument as a string and prints to STDOUT.

How it works

from random import*  Import everything from the random module
def f(x):            Function with input string x
l=10**len(x)         Define l for later use as 10^length(x)
randint(0,l-1)+l     Generate a random integer in the range [0, l-1] and add l, giving a
                     number with l+1 digits...
k=str(...)[1:]       ...convert to a string and remove the first character, giving a key of
                     length l that can include leading zeroes, and store in k
for i,j in zip(x,k)  For each character pair i,j in x and k:
chr(ord(i)+int(j))    Find the UTF-8 code-point (same as ASCII for the ASCII characters),
                      add the relevant key digit and convert back to character
''.join(...)         Concatenate the characters of the ciphertext
print(...+'\n'+k)    Add newline and key, then print to STDOUT

Try it on Ideone

Jelly, 12 9 bytes

⁵ṁX€’Ṅ+OỌ

Try it online!

How it works

⁵ṁX€’Ṅ+OỌ  Main link. Argument: s (string)

⁵             Set the return value to 10.
 ṁ            Mold; create an array of 10's with the length of s.
  X€          Pseudo-randomly pick a integer between 1 and 10, for each 10.
    ’         Decrement, so the integers fall in the range [0, ..., 9].
     Ṅ        Print the key, as an array, followed by a linefeed.
      +O      Add the integers to the ordinals (code points) of s.
        Ọ     Unordinal; convert back to characters.

Pyth - 16 bytes

Waiting for decision by OP on the output formats.

sCM+VCMQKmOTQjkK

Test Suite.