I will be in a very similar situation to you in that my work will be via the Internet from my home in Corfu.
Assuming you will be resident in Greece then your entire income must be declared to the relevant Greek Tax office and you will have to have a greek Tax code and complete a tax return each year (1 Jan to 31 Dec).
If your Internet business means that you are self-employed the legislation requires you to keep ledgers of all income and expenditure, just like in the UK.
If you want to talk about it some more, drop me a private mail to rob DOT sherratt AT gmail DOT com. (replace DOT with . and AT with @, no spaces).