If you're building MVP because you're following lean startup principles, then the next step is to measure. Build > Measure > Learn is the continuous cycle. So start measuring, so that you can learn, and then build based on those learnings, so that you can start measuring...
What should you measure? That rather depends on what your product is - but users, customers, usage, (what gets used, how much, by who, for what).
Of course, you should also continue to bring in new users/customers, measuring and learning how to improve the flow into and through the sales funnel.
If this doesn't make any sense, the next thing you should do is read The Lean Startup by Eric Ries.
After you launch your MVP you need to bring users to it, so you can start collecting feedback. At this stage you are not only testing your product (how it's used), but effectiveness of your marketing channels and value proposition itself. Measuring and learning with a small number of users using your site analytics is very hard, unless you are talking to them directly. You also want to test with a variety of users, so you learnings are very relevant and don't exclude your potential customer group you might not have thought about.
You need to pick a few of these and work on them in parallel:
1. If you didn't get any media coverage at launch and have minimal traffic, try Google AdWords at $5-10/day. It might not be the best ROI long term, but you will learn what keywords and/or product descriptions result in a click through and/or conversions. Setup various goals (email sign ups, registrations, getting to landing page, etc.)
2. Build up your social network profiles by posting interesting content, following other accounts, and participate in discussions via comments. That will allow you to grow your following overtime.
3. Submit your site to directories, deal sites, relevant listings, blogs for review, etc. Just be on the lookout for opportunities for others to mention it to relevant audience.
4. If your product is still free and will be for a while, reaching out to various sites, especially those that list similar tools/resources in your niche and telling them about your product describing the problem it solves could land you some links to your site.
Measure traction at every step in your funnel, and improve any pages where you think the drop off is too high. Just make sure to use meaningful length of time (1-2 weeks) to gather data before introducing a change and keep an eye on external factors (seasonality, spike from a given channel, etc).
At this stage your goal is to maximize learning, so not everything will seem efficient and that's Ok as long as you learn and have new theories to try and test.