Bioships - biological spaceships - appear in a lot of SF, so I thought I'd have a go. However, in most books they are depicted as more powerful than conventional tech in all and any regard. This is unlikely as any given biological system can be outperformed by a mechanical or electrical one. However, if they can be 'grown' without much input then the number in a fleet is massively increased over the more effective but more expensive mechanical ships.
As I'm trying to go for a more scientifically accurate approach several problems have arisen. The most prominent is the nature of the drive system used, both for the ship and for weapons (missiles and torpedoes). So the questions is what propulsion system could a bioship use, based on current understanding, but not necessarily on current ability. In other words, it must not be impossible according to any known science, but very unlikely is fine.
It has been pointed out in the comments that the 'terrain' the bioship operates in has a big impact, as does the exact requirement of performance. It will be mainly within the moon/ring systems of gas giants, along with some(rare) interplanetary travel. Definitely no need for interstellar velocities. Also, low DeltaV hohmann transfers are fine. High acceleration is good but not essential.
All the answers people have given are helpful and complementary, so don't be offended if I didn't pick yours. From the information from this question, along with stuff from around the internet, I have put together some stuff here as a look at bioships for anyone interested.