What makes a good freelance copywriter is the ability to relate to and engage the person who is reading your work. To be able to write in a way which captures the attention and the imagination of the reader (whether your audience is just one, or one hundred) is a very important aspect of succeeded in freelancing.
A good understanding of grammar and English, and the ability to write in a way that is simple to understand, are both essential if you really want to become a freelance copywriter, along with the ability to come up with fresh, exciting copy at a moment's notice. Don't be repetitive (for example, using 'ability' twice in the same sentence as I just did, is probably a bad idea), and get used to proofreading your work before you send the final version to clients.
A good writer who wishes to freelance must learn to adapt and be flexible to new situations and opportunities open to them, which could require them to write unexpected pieces of work on such subjects as "What Makes a Good Writer?". These must be taken in the writer's stride and used to showcase the writer's talents; and not to be treated like a chore.
Dealing with Clients
When it comes to the clients themselves, you have to be flexible and understanding of their needs - for instance, if you're given a set of vague instructions and a 24-hour deadline, don't just try to guess what the client wants; ask them.
Then there is the case of an unsatisfied client - how you react in these situations will affect your future work, and as you'll come to learn, clients are a valuable commodity; without clients, there is no work. Bear that in mind when you address a client's complaints or concerns about your work, and try to maintain a professional level of conduct at all times. Of course, professionalism is a two-way street, so always treat clients (and anyone) in a way that you'd wish to be treated in return.
As someone who has been freelancing for over three years now, I've come to write about all manner of subjects, and thrive on the variety - I think that's fairly important for anyone to bear in mind, especially when times and hard and you cannot pick and choose your jobs. It's great for keeping you on your toes, too, as you never really know what's round the corner.
Be aware of phrasing if writing marketing copy - for example, recently I was asked to re-write a sample copy for a business throwing Christmas parties. They had flyers for both children's Christmas parties, and 'adult' Christmas parties - surely it would have been better to just write 'Christmas parties' or at the very least, 'Corporate Christmas parties', as 'adult' just sounded a bit x-rated! But the flyers had already gone out - a valuable lesson, but at least it provided me with work!
My Final Advice
The best advice I can give for anyone who wishes to become a freelance copywriter is; take your opportunities wherever you can. You never know what situation you'll be in tomorrow, or next week, and taking as many opportunities as possible will help build your reputation and expand your potential. And finally, good luck!