Will CobiT help me with cloud-computing?

Yes, certainly CobiT can help with cloud-computing.

CobiT is an IT governance framework and particularly two processes "Define a Strategic IT Plan (PO1)" and "Manage Third-party Services (DS2)" will give guidance on what you should do before and after selecting a cloud solution, how you could measure and monitor the value you gain as well as your risk exposure and who should be responsible, accountable, consulted and informed during the effort.

Would I really need CobiT to use cloud-computing?

No, not necessarily - but CobiT do provide a maturity model which you could use to assess your current maturity with these processes. If your current non-CobiT-based processes are sufficiently mature you don't have to change them.

If the maturity is not at an acceptable level you will together with CobiT find guidance on how to implement relevant CobiT processes as well as guidance on how to assure that they are implemented and used.