Weight based shipping is calculated using Geo-Zones. You must set up at least one Geo Zone to use weight based shipping. You also need to set the weight for all of your products. You can do this by going to Catalog -> Products, and clicking on the "Advanced" tab when editing a product.
To edit weight based shipping, navigate to Settings -> Shipping and then click the "Edit" link next to Weight Based Shipping.
To enter your shipping rates, you will use the format weight:cost using a comma as a separator. for example:
In this example, anything from 1 lbs/kg or less(depending on how you enter the weight for your products) will be charged $1. Anything 1.01-2 lbs/kg will be charged $3.50, anything 2.01-5 lbs/kg will be charged $7, and anything 5.01-20 lbs/kg will be charged $10. Note that any orders over 20 lbs/kg do not have a price entered, thus would not qualify for weight based shipping. If you would like to use weight based for all orders, be sure to include a very high upper weight bracket.