An RSA public key is comprised of a modulus, n, and an encryption exponent, e.
Therefore you know that n = 91 and e = 5.
In RSA, n is the product of two distinct primes. Once you know these primes, it is very easy to calculate the private key. The particular n you have supplied is extremely small and therefore easy to factor. In practice, n is significantly larger and therefore not easy to factor using traditional methods.
The factors of n are called p and q by convention. In this case p = 13 and q = 7 because p * q = 91.
In order to determine the private key, for classical RSA, you must first calculate the Euler Phi Totient of n:
phi(p*q) = (p-1) * (q-1). In this case phi(91) = 12 * 6 = 72.
Now you must find a value, d, such that e*d == 1 (mod phi(n)) where "==" denotes congruence. In this scenario you are seeking d such that 5*d == 1 (mod 72)
There are many approaches for solving this congruence. The Extended Euclidean Algorithm is a common choice (see your textbook or Wikipedia for an overview of how the algorithm works)
When the algorithm terminates you will discover that d = 29. Therefore the private key is (91, 29).
The encryption function for RSA is c = m^e (mod n) where:
c = ciphertext
m = [plaintext] message
e = encryption exponent
n = modulus
The first value to encrypt:
c = m^e (mod n) = 5^5 (mod 91) = 31
The second value to encrypt:
c = m^e (mod n) = 20^5 (mod 91) = 76
Therefore, the encrypted message would be: [31, 76]
Good luck!