What you want is a VPS, not a cloud. When you start having so much processing and people accessing your website, then you'd want a cloud. For what you describe, just a server where you can run processes on the net is enough.
VPSes are shared between many people so it can be really cheap. Some go as low as $4/mo.
As for domain names, you'd have to purchase one, I like GoDaddy, but you have 1,000's of places where you can get your domain name. Also a place like GoDaddy would give you access to a server too. Just make sure to turn off the FTP access if you are to use that (anywhere!), that's totally not secure. Use SSH with a key.
Can you please emphasize? Why do you think it won't work as is?
Unless your process can be duplicateduplicated on each computer of a cloud and produce the output as expected, it is very likely that you'll need a source of data (a database, a Cassandra cluster, etc.) which you need to access dynamically. This source of data needs to be elastic as well because otherwise increasing the number of front end computers will generally be useless.
So... say you create a web page in perl, that perl script will need to appear on each cloud computer. That perl script needs data, it has to query a computer for said data. If that computer is the only one answering to all the front end web pages, you have not resolve the problem: 1 million hits per second on the backend will kill the backend.
So... now you have to duplicate the perl script + the data + keep the data in sync. on all the computers forming your cloud. That's not something you do overnight... You have to remember that a cloud is formed of distinct computers, not just additional processors, NICs, hard drives, etc. in one computer.
If you're interested about such things, the Cassandra website has interesting documentation in that regard.