hello!
interwound world!
firstly you will need to satisfy the EU guidelines for billing EU companies for professional services rendered, from extra EU countries, e.g., itemized bills, etc
secondly, on the same lines, you will need to verify THERE whether you need to add VAT - typically if billing from the USA, no
The most linear part is actually how you treat your income in the USA - you will treat it as, if that is what it is, income from professional services rendered. Where you will declare this income will depend on how you bill for it - if as an individual in sch. C, and so on
So firstly verify with your client what are their billing requirements, VAT, etc.
Then report your foreign income in the US in essence as if it was local income -careful with other reporting requirements such as declaring foreign bank accts to FINCEN, etc., Self-employment tax if that is the way you are going to bill, etc. Keep track of ay foreign tax paid or expense in which you incur since they may become tax credits or deductions.
In general, IRS penalties for willful under reporting of foreign income/accts has extremely severe associated penalties.
I highly recommend you seek local professional guidance with a person versed in foreign income rules & regs to be educated properly on how to set up an appropriate business entity & the bookkeeping for the same. Doing things incorrectly can cost you thousands in penalties later so a few dollars spent on education now is well worth it.