@PinkMonkeyBird
Welcome to Battlefield!!!
Battlefield is launched thru the site called Battlelog. I believe your problem is that you are using the Chrome browser. I love Chrome, and used to use it for launching the game, but a few months ago, Chrome dropped support for the Plug-in that is used for launching the game. They have released a "Plug-in Free Launching" system, but it doesn't work for my friends and I. It works for some, but we have not gotten it to work. I would start off by using Firefox (this is the browser I use to launch the game). Sign into your Battlelog there. Then hover over your name in the top-right corner of the page, and click "Settings". The top option is "Game Launch" and this "Use plug-in free game launching" is ON by default. Click the slide bar next to this and turn it OFF. Then to the left of your name, you will see a Yellow circle with an arrow in it. Click this and download and install the plug-in. After this, Reboot your PC to ensure it has installed fully. Then sign back in, and try to launch the game now.
Post back here and let me know if you were able to get it to launch, or if you have any further questions.