This achievement, along with all online achievements for this game, is now unobtainable. The multiplayer servers have officially been truned off as of August 11, 2011
Not all kits have grenades, so pick either Assault **(The underslung M203 grenade launcher aka "Noob Tube" also counts as a grenade kill)**, Support, or Engineer.
After you reach the required 20 kills you'll hear a "Buh-dum" sound followed by a message that will pop up on your screen notifying you that you've be awarded the Grenadier Medal. No achievement quite yet, just keep playing until the round is over. To get the achievement to pop simply join up another game and it should pop once you have loaded into the match. I personally had problems getting the achievement to pop if I left or got disconnected from the match where I had met the requirements. If that happens you will need to go through the steps again.
The easiest way to obtain this (or any MP achievement) is to obviously boost for it. If going this route, your best bet is to try the Clan Match method with at least 1 other person to prevent randoms from messing things up. To do this, 2 people need to create a clan within the multiplayer menu. After that you can invite other players to your clan (if you have a group larger than 2 people) via the guide button while in the game. While in the main menus of the game press the guide button, click on the profile of the gamer you want to invite to your clan, then click on the "Opt for Battlefield Clan" (not sure about the exact wording) option.
After the clans are set up both Clan leaders (the people whom created the clans) need to ready their clan through the "Clan Games" option. Once the clans are ready 1 clan leader will need to search for available clan games via the "Clan Games" option. By selecting that you will be brought to a Server list where you should see the name of you boosting partner's clan. Highlight the desired clan and select it. You will then be brought to a menu where you select the match options. Leave everything to "Any" and only change the mode to "Conquest" and the map to "Damage" (Easiest map to end games with by controlling flags and making it impossible for the opposing clan to spawn, thus ending the game instantly). After selecting your options press "A" and wait. Your boosting partner should see a clan message notification at the bottom of their screen which they access by pressing "Y". Once they accept it the match will be loaded and the leader will automatically join. Any other clan member will automatically receive a game invite that they can accept or they can join through the "My Clan Games" menu.
Once in the game, the Achiever(s) should make their way over to the enemy spawn which is indicated on the radar (top right hand corner of the screen) as a red arrow. When everybody is in place do your thing and get some kills! Make sure nobody captures the flag as the match will end prematurely! If ammo becomes a problem, note that there is an Ammo box at every flag where you can take care of that.
**Credit goes to Jaggexedge for commenting on that addition**