Technically, all DNA is digital already.
Digital, in it's strict meaning, just means that the data contained in the DNA can be stored in discrete packets and that it is not based on a continuum of (say) a wave function, like sound.
Digital music is different to analogue music in that sound is actually a vibration, or a wave. Sound as we understand it is strictly analogue. BUT, since CDs and MP3s and the like, we have had the ability to slice the waves into sufficiently small enough chunks that we can record the sound in discrete packets, and replay it in a way that sounds VERY close to the original.
Is it perfect? Well all those people going back to Vinyl records will tell you no. But, for most people's needs, it's close enough.
DNA though is different. It's not a waveform, it's not analogue. It's encoded in discrete chemical packets. DNA is actually written as a sequence encoded using 4 'letters'; A, G, C, T. As such, it is entirely possible to record a DNA sequence digitally and perfectly. It's just very long, and then recreating the DNA chain is an engineering feat in and of itself.
So, what would a creature with digital DNA look like? It would look like us. (Not human of course, just biological). Could we clone an alien? Theoretically yes, but there is a catch. Even if we master the cloning of terrestrial life forms, even humans, that doesn't mean we can use the same technology to clone aliens right out of the bat.
Why? Well for one thing even though aliens are likely to have a chemical (and therefore digital) DNA construct, it's just as likely that they'll have a different 'alphabet', or series of chemical modules that string out to form a DNA chain. So, in order for us to be able to clone an alien, we would first have to know their DNA coding, understand the chemical chains and the 'letters' that form their DNA alphabet, and then know how to recreate them.
But the key point is that DNA is already digital in the strictest sense. Storing it as a data sequence via a magnetic sequence of 1s and 0s is the easy bit; Knowing how to read those 1s and 0s back into a chemical processor that can recreate the correct DNA molecule strain is the hard bit, but ironically enough, it's going to be easier to do that perfectly than it ever will be to perfectly recreate a song in a digital format, no matter how close we get to that.