Yes, there are multiple flights from Bangkok to Genoa for under ฿ 23,956. The cheapest flight booked recently is on KLM for just ฿ 21,321, but on average you can expect to pay ฿ 28,987.
There are no direct flights from Bangkok to Genoa, but Cheapflights can provide you with all the information you need to find the fastest route with the fewest stops.
You can imagine temperatures to be in the 36 °C range in April. The weather in Genoa is generally colder than Bangkok's. As flight availability is subject to become more limited as peak season approaches it is ideal to book in a timely manner.
Lufthansa, Etihad Airways, and KLM can get you from Bangkok to Genoa in just under 15h 45m. Your travel time can differ by up to 14h 35m. Both multi-stop and direct flights can be found to your destination and back.
Flying on KLM is currently your cheapest option to Genoa, with prices starting at ฿ 21,321. Additional return tickets that might be worth considering are offered by Etihad Airways and เอธิโอเปียแอร์, with the lowest prices starting at ฿ 21,561 and ฿ 24,975, respectively.