In my up coming novel this fall, I got the human crews as well as some humanoid alien species working and living onboard a FTL starship exploring an uncharted cluster of neutron stars and black holes. As the story progresses the starship will visit strange and unfamiliar worlds and add diversity to the existing crews, I'm seeking a carbon based living organism which doesn't share our cell structure or DNA for reproduction and I also need the unique environment to sustain such a lifeform. Last but not least how can it stay onboard the vessel without jeopardizing its life?
The only known non-cellular lifeforms are the viruses.
A virus is a small infectious agent that replicates only inside the living cells of other organisms. Viruses can infect all types of life forms, from animals and plants to microorganisms, including bacteria and archaea.
This suggests the most probable environment where a virus can exist, is inside the living cells of other organisms. This might be extended to some sort of generalized organic system. OK. A mass of organic or biological material, possibly undifferentiated, rather like a living blob or perhaps an organic pool.
The origins of viruses in the evolutionary history of life are unclear: some may have evolved from plasmids—pieces of DNA that can move between cells—while others may have evolved from bacteria. In evolution, viruses are an important means of horizontal gene transfer, which increases genetic diversity. Viruses are considered by some to be a life form, because they carry genetic material, reproduce, and evolve through natural selection. However they lack key characteristics (such as cell structure) that are generally considered necessary to count as life. Because they possess some but not all such qualities, viruses have been described as "organisms at the edge of life", and as replicators.
Source for the above quotations is the Wikipedia entry on viruses.
The OP has been given references to sources of information about alternatives to DNA, in the comments above, these do not need to be duplicated in this answer.