This quest should not be too difficult. Rare weapons are quite good so eliminations can come without much difficulty. This quest includes all weapons of any class. If you have a blue coloured it will count.

If you want to get this quest done quickly you can jump into a team rumble match. Try and get yourself an assault rifle that is at least rare. There are plenty of supply drops in this game mode so maybe you can get something better.

When the storm circle is small near the end of the game it should be easy to get up high and pick off enemy players. If you are eliminated you keep your weapons so don't be afraid to take a few risks.

Depending on your play style you might prefer to go in up close and personal so a rare or higher shotgun or submachine gun will work just as well.