There aren’t any direct buses or trains from Edinburgh Airport to Dundee. Any journeys will involve at least one change somewhere.
For travelling by bus, the quickest and cheapest option would probably be via Halbeath park and ride, near Dunfermline. This is doable in about 1 hour 40 minutes, depending on what time of day. From Edinburgh Airport, take the Stagecoach Jet 747 service to Halbeath, this runs every 30 minutes. Then you can get a Megabus service from there to Dundee. You can book a ticket for this whole journey, from either Citylink or Megabus. Megabus is usually cheaper, though the ticket will be for a specific journey. Or you can buy a ticket from the bus driver, though this would be more expensive.
Alternatively, you could go via Edinburgh city centre. Take the Airlink bus service 100 to Waverley, then it is a short walk to the bus station on St Andrews Square. The Airlink bus runs every 30 minutes. Or you can get a tram, these are more frequent, but may take a bit longer. Then you can get a Citylink or Megabus service from Edinburgh to Dundee.
Or you can go by bus or tram into Edinburgh, then a train to Dundee. This would be quicker than a bus, but probably a bit more expensive. For this, get the Airlink bus or tram to Haymarket station, where you can catch a train to Dundee.
To check times for any of these services, you could use the Traveline Scotland website. This includes nearly all buses, tram and train services in Scotland. You can search for journeys on a specific date and time.