I am pretty sure that s3 only allows for static hosting and Amazon has a relational database that you can use; however it isn't mysql.
The database costs about $0.11 an hour. It would take some work to have your site hosted by s3.
There's plenty of other choices out there which are better/easier than Amazon.
Amazon is more set up as "Cloud Computing", whereas you are probably looking for a "VPS" or similar service. Cloud computing is about dynamically allocating resources, scaling quickly and using only the capacity you need. Paying by the hour is ideal when you only use the extra resources for a relatively short period of time, not for always-on services.
IMO, Amazon is really only good for very large sites which need to grow/shrink dynamically based on load (eg, seasonal traffic surges such as Christmas shopping, etc). Otherwise, it's just too much work.
Try sites like Linode for always-on VPS hosting. Lots of other VPS choices out there too.
As Sim said, VPS is probably the way to go for you.
Can highly recommend nimbus for forum hosting, they are currently doing an excellent job with our large (and resource hungry) VB4 forum, which will be converted to XenForo as soon as 1.1 is released properly and there is fully working VB4 importer.
The xen test forum I've setup (imported over 2 million posts!) with them absolutelyflys!