If you want to travel from Bangkok to Koh Lipe there are a number of ways you can get there and as there is no airport on the island, you will have to at least do the last leg of the journey by ferry (high speed boats).
The best way to travel from Bangkok to Koh Lipe is via Hat Yai with many boats operating year round (weather permitting) from Pak Bara which is easily reached by minivan.
Another option is to travel via Trang, but you should only do this during the high season (November to March) as boats do not operate at other times of the year.
The nearest airport in Thailand to Koh Lipe is either Trang Airport or Hat Yai Airport.
You can get reasonable cheap flights from Bangkok to Hat Yai using Thai Air Asia, Thai Lion Air or Nok Air from Don Mueang Airport or Thai Smile from Suvarnabhumi Airport.
From Bangkok to Trang, there are less flights with just Thai AirAsia and Nok Air flying from Don Mueang Airport.
Aim to get an earliest flight so you can get to Koh Lipe on the same day.
If you decide to fly to Hat Yai (the most popular and travelled route), you can take a taxi or cheaper minivan (minibus) directly from Hat Yai Airport to Pak Bara pier in Satun, which is the main port for boats out to the island (should take less then two hours to get there - about 120 kms).
From Trang Airport you can also take a taxi or minibus, but to Hat Yao Pier.
It's only a short 50km drive, but the ferry takes almost twice as long and there are far fewer boats.
From Pak Bara to Koh Lipe there are a number of high-speed boat operators that operate many times a day out to the island all year round (weather permitting), taking around one and a half hours to make the trip.
Taking an overnight train from Bangkok to Hat Yai or Trang is a good alternative to flying and also cheaper.
You can get a comfortable sleeping berth or if you really want to save money, a seat on one of the slower Rapid Trains that are really cheap.
There are a number of night trains that leave Bangkok Hua Lamphong Railway Station every day, arriving in Hat Yai Railway Station the next morning, less trains to Trang (which is on a different branch line).
From just in front of the Hat Yai station or Trang Station you can than get minibus transfers to the ports, then the boat to Koh Lipe.
As with many popular destinations in Thailand, you can almost always find a Khao San Road bus company that operate on the route.
Jolly Travel operate a bus and ferry service from the Khao San Road to Koh Lipe.
It's an over night service leaving in the afternoon and arriving in Koh Lipe by lunch the next day.
If you want to take a more luxurious bus to Koh Lipe, take one of the overnight VIP buses to Hat Yai or Trang from the Southern Bus Terminal in Bangkok, then transfer by minibus to the ports.
If you are travelling from other parts of Thailand's south-east coast (Koh Samui, Koh Phangan, Surat Thani, Koh Tao, Chumphon, Hua Hin etc) to Koh Lipe, just follow the same travel routes as per above via Hat Yai (easiest) or Trang.