You can definitely use one app for leads and customers, and distinguish between the two using a category field.
I generally create a "Contacts" app for this with a relationship field (values of "Lead", "Customer" for example), and then have a separate "Customer Orgs" app. That way I can add a relationship field on the Contacts app that references the Contact's Customer Organization. And you can easily keep track of the Contacts per Customer Organization this way.
Globiflow (included with Podio Premium if you pay for 5 users) is likely to be a big help to you at some point.