To see a wider selection of flights to Kiev, try searching by city rather than by airport.
If you don’t have specific dates for when you'd like to fly to Kiev, try searching by month. Flights are usually cheapest in June, as this is low season in Ukraine. It's also worth looking out for flights with SWISS, as their fares are normally the best-value.
June will generally be the most expensive month to go to Kiev, as it's peak season. Accommodation is typically pricier at this time of year too. If you want to go at this time of year, try looking at other cities in Ukraine to get a cheaper flight.
Whoever you want to fly with, it's best to book your flights in advance, particularly if you want to fly on a specific date – otherwise they may sell out. Once you've booked, why not take a look at accommodation? Simply click on the ‘Hotel’ tab at the top of the page.