Use the server browser to find a Conquest game and try to find a game that is half full. The map Sinai Desert is the easiest for this strategy.
Choose a game of Conquest because it is the easiest to level up in given the game's length and the scores it provides. It is the biggest in Battlefield 1's multiplayer modes.
Don't select any of the available Squads. Create your own Squad to be the Team Leader.
Choose the Class you want to level up. While doing so, make sure you're leveling up the actual Class you want because spawning into a Tank will cause you to rank up that class. That also means that you cannot spawn in a vehicle if you want to level up your Scout Class.
Mark an objective (flag) while you're a squad leader in order to gain extra points upon capturing it.
Capture the flag. You gain 25 points for each 5 seconds you stay on the flag, 250 for neutralizing it (45 seconds) and 500 for capturing it. That's a total of around 1175 points without the Squad Leader bonus.
You can use one of the vehicles available to move faster. The recommended vehicles for this are the Armored Truck and the Motorcycle. However, DO NOT spawn directly into them unless you want to level their classes up. You can also use a horse but don't spawn as Cavalry.
Multiplier Boosts can extend the leveling up stage drastically if you play well too.
You can level up around one level per match. Use your Companion App to keep track of your level and unlock the level 10 weapons.
Don't pay attention to your kill/death ratio. Each kill equals 100 points while capturing a flag is equivalent to around 1175 points.
Battlefield 1 is a game that requires strategic thinking. In a way, sometimes the best solution to level up is to camp in a place far from enemy reach where you can see them coming.
Capture the flags nearest to the spawn-base first. The first two are usually the easiest to capture.
Use strategy to capture the flags around you. If you see a flag being captured by enemy troops, stay away from it and move to another one of their flags that has been abandoned. Players usually wait a few moments before moving to the next flag post. Capturing abandoned flags is easy.
There are currently ten maps for the Conquest game-mode in Battlefield 1. You must choose the class you want to level up according to the map.