In my recent experiences with running this game, it's been a doozy of a time trying to get this game to capture. To save you some time and effort, allow me a couple minutes to give you some instruction for how to capture your records.

Firstly, go ahead and download OBS. OBS is a software available for FREE to help people like you and me Stream and Record gameplay at a far easier rate than other services cough cough Shadowplay... Aside from that, setting up what you want to record and how you want it to be recorded are far easier. When running OBS, set your scene; comprise it of Game Capture (set to the window of the game you're playing) and an Audio Output (either Stereo Mix, headphones, whatever you have hooked up). All you need then is to start the game.

Secondly, ASSUMING you have installed the modern patch for NFS3 (or installed the already torrented game) you would then need to press Alt+Enter to run the game in windowed version to allow for the recording software to capture your screen. Fear not, the menu will appear small, but the game will be ran in fullscreen.

Thirdly, upon completing your run with any records you broke, SAVE THE VIDEO. The number of times it pays to have original footage saves efforts on so many people. Edit what you need to (trim video to approx. 5 seconds before the lap begins and approx. 5 seconds after the lap is completed) and post it to YouTube for safe keeping. Since ANYONE can make a YouTube account, it should be pretty straight forward and simple to make an account and begin uploading. Remember, your video should include the following.

 A. Proof of lap time completion
 B. Information of track, conditions, etc.
 C. (Optional) Provide details on tuning such as a screenshot