What all clients want
Each client is different - in the way they see your role, in the level of involvement they wish, and in the sort of results they seek.
With experience, one learns to read the situation, adapt, and establish an agreed working relationship. However, the following are common to virtually all clients:
No Unexpected Surprises
More anything else, this needs to be the touchstone that governs any consultancy assignment. If anything is going amiss, clients need to know. In particular in case it impacts on something else they or others are doing.
You want success but so do your clients - it reflects on them just as much as it reflects on you. But how does your client define success? Spend some time in the early days establishing just that.
Clarity and Flexibility
They want to be clear what your role is and what you are planning to do, but also expect you to be flexible and able to adapt to changed circumstances. Thus, work to define your role and then stay in it, but be prepared to flex and adapt when needed.
Your Interest in them
They expect you to be interested in their work, their ideas, their people, and the issues they face. Try to exhibit that.