Willingness to keep educating yourself
As stated on Brian Tracy's blog, career counselling is constantly evolving. New trends, new words, and new practices come and go every year. A professional coach has to keep pace with all these trends, which is why you have to be prepared to educate yourself continuously.
The ability to listen and empathise with people you are talking to is a prerequisite for becoming a career coach. Learn to see things not just through your own eyes, but also through the eyes of those around you.
Planning and organising your work
A career coach needs to be well organised. You have to come across as professional, meet all deadlines and fulfil all promises; you also need to be able to organise your work excellently, as well as teach your clients to do the same.
One of the basic roles of a career coach is to motivate and inspire clients. You should therefore have a generally positive (though still realistic) view of the world and be able to motivate your clients to view the world as a set of opportunities.
Ability to challenge a client's beliefs
Clients will often have completely misguided but deeply ingrained ideas about how the job market works or where their own strengths and weaknesses lie. You must be prepared to some extent to challenge these beliefs and set them straight, even at the expense of an argument or mild confrontation with the given client.