I think you can do this for a reasonable price, depending on dates and how far in advance you book it, with one-way tickets you book yourself. Be prepared to spend some time on the internet looking at the relevant airline sites. My suggestions below are based on airlines flying to main airports that sell one-way tickets for a sensible price (some airlines charge very high fares for one-way tickets):
London to Paris: BA from Heathrow sell one-way tickets, and are not necessarily that expensive, and you can check in a suitcase for free. They have flights to both main Paris airports, CDG and ORY. Or easyJet from Luton. Unfortunately there aren't any flights from Gatwick (to tie up nicely with getting easyJet back to Gatwick for the last leg if you are parking your car at the airport or buying a return train ticket to the airport).
Paris to Barcelona: Vueling is an established Spanish low cost carrier.
Barcelona to Rome: Vueling again (random day in June I just looked at had seats at 19 euros!)
Rome to Athens: Aegean Airlines a Greek airline are selling one way tickets at under 100 euros. And easyJet fly the route too.
Athens to London: BA or Aegean to Heathrow, or easyJet to Gatwick
Izmir to London: easyJet to Gatwick
Make sure that if you are buying separate tickets for each flight that you allow plenty of time in between flights so that if you have a flight delay you have some contingency - if you are on separate tickets the airline won't have responsibility to get you to your next flight on time. I am guessing this won't be an issue for you because it looks like you are stopping in each place.
And with so many flights be very careful about your dates, name spellings etc when booking flights on the internet.
Goes off to ponder new career as travel agent....