To become a copy editor you would usually need a degree and experience in publishing at an assistant level.
Degrees in publishing are available at a number of colleges and universities – visit the Publishers Association website for details.
To work in a specialist area you would find it useful to have a relevant degree. For example a science degree would be useful for scientific publishing. After completing a degree, you would usually need to start in a position such as editorial assistant.
You may be more likely to find work with small, less well-known publishers to start with. This would help you to get some experience of editing, build up contacts and develop a portfolio, so that you would be more likely to be considered by larger publishing companies.
Joining professional organisations such as the Society for Editors and Proofreaders (SfEP) and Women in Publishing would be useful, as it would give you opportunities for professional support and networking.
You may be able to find freelance work through contacts you have made in the industry, or by making speculative approaches to publishing companies.
You can find contact details on the members' lists of organisations such as the Publishers Association. Some of the larger publishing houses run graduate training schemes.