I need the shipping fee calculated according to weight on my Wordpress website

General information for the website: We sell eco friendly disposable cutlery/packaging
Kind of development: Customization of existing website
Description of requirements/features: I have been losing out on shipping fees as i've been charging a flat rate for shipping regardless of whether the customer buys 1 product or 10. I have now added weights to each variable, so what I need is for the website to add the weights of all the products the customer is buying and then allocate a shipping fee.

Currently I charge the following for shipping:
Flat rate: £3.90
Signed for: £4.30
Free shipping (orders over £30)
(This is regardless of whether 1 item or 10 items are bought)

I would now like to move to the following shipping fee structure:

Weight (kg) Cost
between 0 - 1 £3.78
between 1 - 2 £4.78
between 2 - 5 £7.79
between 5 -10 £10.49
between 10- 15 £13.78

Website:
http://www.greenspoons.co.uk (Wordpress)
Extra notes: