Read a lot and pay attention to fine details. You can read books, articles online or the newspapers and look for errors, even minor in nature. This will give you a chance to develop your skills. Also mark these errors when you find them.
There are many institutions that offer classes on proofreading and it is not a bad idea to join a class and learn the trade from the professionals. This will also give you a formal training on how to do proofreading on professional lines and you will be able to build up on your previous efforts.
Start writing and provide your work to local newspapers and even write a blog. Offer your proofreading services to the newspapers and magazines as well as be willing to help them in other departments as well. If they do not offer you any monetary benefits in the start, that is fine. You are just gaining much needed experience in the field.
You can also help blog writers with proofreading their work. Moderating forums is not a bad idea too as the work is similar in nature.
Make a Resume
Once you have gained some experience, make a resume that focuses on the abilities you have to write, edit and proofread. This is important as these are the areas in which you are looking to land a job in. Make sure that you make a clear mention of all your experiences.
Start working freelance, there are many websites that offer opportunities and this will greatly enhance your experience. It will also reflect well on the resume and you will be earning a decent amount of money at the same time.
Make sure that you keep on applying for a full time position with publishing houses. With time you should start getting offers and will be able to find a job of your liking.