The title of your thread says you are selling a website, that is, one that already exists. But the rest of the question suggests that the client is paying you to create a new website. Perhaps you could clarify which of this it is.
Assuming we are talking about creating a new site, that is, a custom site for this specific client, then the steps you listed are approximately correct, except that you missed out something very important:
- Client pays the remaining 50%.
But there's more to it than that.
You need to agree with the client the basis of the charge, that is, whether it will be so-much per hour, or a fixed price, or some combination of those.
If it is an hourly rate, and if the work is likely to take more than a month, I think it's better to bill the client monthly, rather than in two chunks as you suggested. At the end of each month, bill them for the actual hours you spent on the work in that month. But that's just my opinion. Practices vary.
The other important point is to make sure all the details are agreed in writing. You need a piece of paper setting out exactly what the work involves, how much you will charge, how long you will take, what happens if you fail to deliver, how quickly you will be paid, and other similar contractual details. It might be a formal contract, or you might just write to the client with that information, and ask him to write back to say he agrees.
I'm sure other people will offer more advice, so I'll leave it at that for the moment.