There are three budgets run by the Commonwealth: the Commonwealth Secretariat Fund, the Commonwealth Youth Programme (CYP) Fund, and the Commonwealth Fund for Technical Co-operation. (source)
The last of these is the largest (£30 million) and is funded by voluntary contributions by member states, with the UK, Canada, Australia, New Zealand, India, Nigeria, Brunei and Kenya making the largest contribution.
The other funds total about £20 million and are funded based on "ability to pay" as assessed by the Secretariat of the Commonwealth, who take into account such factors as GDP, population and debt burden and any special circumstances affecting that country. The Secretariat seem to have a degree of flexibility, rather than simply applying a formula.
If a country fails to pay, but otherwise fulfils the conditions of membership, it will be classified as a "member in arrears" and have a limited access to the institutions of the Commonwealth.
I was unable to find a list of contributions, and this information may not be in the public domain. However, it is clear that the rich nations contribute considerably more than the poor ones.