The digital signature algorithm encrypts a hash using the senders private key and the receiver's public key.
Huh? I see two problems with the above statement;
"Encryption"; using the word encryption implies that there's a way somehow to decrypt it. However, there's no way to anyone, even with the private key, to "decrypt" a signature to generate the hash.
"using ... the receiver's public key"; the signature operation most certainly does use the signer's private key, but it doesn't use any key (public or private) from the receiver.
On the other hand the RSA-based digital signature algorithm encrypts the hash using only the senders private key, which is a much cheaper operation.
Not necessarily; if we're talking about the standard DSA operation, the expensive operation is a modular exponentiation of the generator over a (perhaps) 256 bit random exponent modulo the prime; for RSA, the expensive operation (I'm assuming CRT) is two modular exponentiations of arbitrary values modulo primes half in length of the key size. I believe that you can implement the DSA operation (which, using precomputed tables, would take perhaps 50 multiplications modulo a 2048 bit prime) considerably faster than you can the RSA operation (which might take perhaps 2000+ multiplications modulo 1024 bit primes)
In addition, with DSA, you can potentially perform this computation before you learn the value you're signing; if you can do this in time where you would otherwise be idle, you can make the signing operation very cheap indeed.
And, if we're talking about DSA over Elliptic Curves, that is, ECDSA, the balance tilts even more radically towards ECDSA.
Where RSA shines is the signature verification operation; there's, it's considerably faster than (EC)DSA.
Under what circumstances would I need to use the expensive DSA as opposed to using the much faster RSA?
Quite apart from your mischaracterization of DSA as expensive and RSA as cheap, actually, straight DSA is fairly rarely used in practice. ECDSA (and relatives, such as EdDSA) are becoming more popular, in part because it is cheaper...