Major international airport of Japan are Narita International Airport(NRT) and Haneda International Airport(HND).
Also Kansai International Airport(KIX) in Kansai area is available.
Although the nearest airport of Kyoto is Osaka International Airport(ITM), it is domestic flights only.

To Kyoto Station

Please refer to the followings from each airport to KYOTO station.

from Narita Airport(NRT)

https://www.narita-airport.jp/en
It is recommended to take domestic flights to Kansai airport (KIX). If you take trains to Kyoto, please refer below:

By Train (JR Narita Exp.)
to Shinagawa Sta. (or Tokyo Sta.) : About 60 min.
Time Table →Click

Tickets Buy: Ticketing machine or JR Ticket Office of Narita Airport Sta.
Terminal 1 sta.
http://www.jreast.co.jp/e/stations/e1130.html
Terminal 2・3(Airport Terminal 2) sta.
http://www.jreast.co.jp/e/stations/e611.html

from Shinagawa Sta.(or Tokyo Sta.)

http://english.jr-central.co.jp/info/station/index.html

By Train (JR Tokaido Shinkansen)
to Kyoto Sta. : About 130 min.
Time Table: →Click

Tickets Buy: JR Ticket Offices or JR TOKAI TOURS
http://english.jr-central.co.jp/tickets/buy/index.html

from Haneda Airport(HND)

http://www.haneda-airport.jp/inter/en/
It is recommended to take domestic flights to Osaka International airport (ITM). If you take trains to Kyoto, please refer below:

By Train (Keikyu Line)
to Shinagawa Sta. : About 11 min.
Time Table: →Click (Route search)

Tickets Buy: Ticketing machine of Keikyu Line.
http://www.haneda-tokyo-access.com/en/transport/trainlinemap.html

from Shinagawa Sta.

http://english.jr-central.co.jp/info/station/index.html

By Train (JR Tokaido Shinkansen)
to Kyoto Sta. : About 130 min.
Time Table: →Click

Tickets Buy: JR Ticket Offices or JR TOKAI TOURS
http://english.jr-central.co.jp/tickets/buy/index.html

from Kansai International Airport(KIX)

http://www.kansai-airport.or.jp/en/index.asp

By Train(JR Airport Exp. Haruka)
to Kyoto Sta.:75 minutes
Time Table: →Click

Tickets Buy: Ticketing machine or JR Ticket Office of Kansai Airport Sta.
http://www.westjr.co.jp/global/en/travel/shopping/access/

By Bus(Airport limousine)
to Kyoto Sta. : about 105 minutes
Time Table & Bus Stop: →Click

Tickets Buy: Ticketing machine of Airport Bus Stop.

from Osaka International Airport(ITM)

http://www.osaka-airport.co.jp/en/

By Bus (Airport limousine)
to Kyoto Sta.: about 60 minutes
Time Table & Bus Stop: →Click

Tickets Buy: Ticketing machine of Airport Bus Stop.