Go into System > Configuration > Shipping methods (on the left hand side).
From those options there enter the Table Results.
Enable this and make sure to set the condition to Weight vs Destination. Name the Title and Method name. Select save config then select your store from the current configuration scope at the top left bar.
Then re-enter the shipping methods and then Table Results for your store. You should see a variety of options. Export the CSV file. Once you open the CSV file you should see options for Total Weight and Shipping Price. From here you can enter the price of shipping specific to the weight. Once done import the CSV file back into the Table Results option page from earlier and then select save config. You should now have table rates.
The following link is a guide that should help you if you get stuck. It is the same method but it discusses how price vs destination works which is just the same as how weight vs destination works, the only change being the options in the CSV file.
If you want free shipping :- Add weight at 0 with shipping price at 0. Then disable free-shipping in the options for shipping methods in your store.