You say in the title you want to travel from Bourgas (in Bulgaria) but in the text you say Bursa - I presume you meant Bursa.
There are a number of different ways, but by far the fastest way is to take the IDO fast ferry from Güzelyalı (about 7 miles from Bursa) to Yenikapı (near the Sultan Ahmet area of Istanbul). The ferry takes 1hr 20mins. There is a municipal bus between the main bus station in Bursa and the ferry port. It has only just started so many people you ask may not know about it. Follow this link for details http://www.ido.com.tr/en/index.cfm?page ... =253&ln=en
Alternatively, you can take a bus to Yalova (50mins-1hr) and take the ferry from there to Yenikapı, which takes 1hr 10mins. If you are going to the Asian side of Istanbul, the best option would be to go to Yalova and take the fast catamaran to Kartal (40 minutes) or failing that, the car ferry to Pendik (45 minutes).
Finally, there are many direct busses travelling between Bursa and Istanbul (practically all busses on their way from Istanbul to the Aegean region stop off in Bursa on the way), but the land distance is much longer than the sea distance and they will take around 4 hours (potentially much longer, if there is traffic) to get to the European side of Istanbul and then you'll have to get into the city centre.
There is no train service in Bursa except for the metro system.
Timings for the ferries are available on the IDO website; http://www.ido.com.tr/en/index.cfm
Bear in mind that the Yalova-Kartal route is a "inner-city" journey, but all others are "inter-city".