Personalised champagne is quite a broad term, so let's start by defining what we don't mean.
I'm not talking about having it blended especially for you! Yes this is possible if you are super rich, but not for us normal folk.
Perrier-Jouet did offer this service to 100 customers last year, but at £38,000 for a case of 12 bottles it was out of most peoples reach.
What we do mean is that the bottle, and more particularly the label is personalised, so that it is effectively unique.
This is a particularly Anglo-Saxon idea.
Most English speaking countries will have this service available, but it is much more limited in mainland Europe.
In some areas this may only be available on personalised sparkling wine rather than proper personalised Champagne, but the bottle contents don't effect the concept.
Personalisation possibilities.

• Personalised neck bands.
  are possible but not widespread.
  There are two key difficulties with this: firstly is that people rarely spot that this has been done, so it is fairly ineffective; and secondly it often doesn't look as good as when the neck band is added on the bottling line.
  
• Engraving on the bottles.
  This can look nice if done properly but it is usually limited to a short phrase or name because the label takes up most of the space on the bottle.
  Don't try it at home with an engraving set as the bottle is highly pressurised, and if you go too deep it can shatter.
  
• Painted bottles.
  These can be very colourful but they don't look like proper champagne, so the impact is reduced.
  
• Personalised labels.
  The most widely offered method, a properly printed label stuck on a bottle - just like they do in the winery.
  

Personalised labels Once again there is a lot of choice in the category.
Labels range from the possible outdated type of just a message on a plain label, through adding your message to a template design themed to your occasion, to even uploading your photo on to the label template.
If you are looking for several dozen or more bottles then it is also possible to have a completely bespoke design.
You may be able use a designer or even use your own design if you have suitable software.
The options will vary between suppliers, but you should have no difficulty finding what you require.
One point to bear in mind is that you must choose a supplier that can ship to your location.
In the USA the interstate alcohol shipping rules are complex and restrictive.
Elsewhere it's the more practical point the bureaucracy (3 copies of the invoice for Customs), and the risk of transporting fragile items.