The most important thing in my mind is that the consultant should let you know everything they are doing without obviously giving too much away.
What I mean by that is they should tell you that they are doing a specific task but they might not want to go into detail of how to perform that task.
Here are most if not all of the tasks that I would expect an SEO consultant to perform for you:
An SEO audit should be the first port of call. It will identify any technical errors with the website such as broken links or missing meta tags.
The next step would be to do some competitor analysis and keyword research. The consultant should then agree with you a list of keywords to work on and monitor.
They should provide you with a report which shows the current rankings of these keywords along with any other KPI you might want to track including website visits, domain rating etc.
If you haven't already got Google Analytics and Google Webmaster Tools set up they should help you to get these created.
The next step would be to optimise your website for the agreed keywords. This will require making sure that you have the right content. You might require additional pages.
Your web pages then need the meta tags and image tags optimising.
The consultant should also look for internal linking opportunities from other pages on your site such as information pages or blog posts.
They might also recommend adding new content on a regular basis.
Backlinks are an important ranking factor and they should be able to assist you in getting backlinks on relevant sites such as local directories and may also suggest getting some PR work done so that you get exposure on journals or news sites.
Monthly reports should be done so that progress can be tracked.