Here's a link to some great videos covering all of the missions...https://www.youtube.com/playlist?list=PLtXbDwXf4tU5ipQE3gS9g...
He will often trounce the times/scores etc but don't worry if you're not so close it's probably worth finishing the race to see if you still get the platinum, for example on the race 'Whiteout' he got a time of 5:01, I got a time of 5:14 and still got the platinum.
Note: Some races have multiple stages, for example a loot pick up followed by a follow section, when you hit options and press restart it will restart the section and not the entire mission which is very useful if things go awry.
When on YouTube remember to hit CTRL+F and type in the race name to find it easier.
I used a RUF3400 K for all specs apart from the street spec for which I used the Skyline. You don't need the best car for each spec but of course it's gunna make things a bit easier.
Feel free to comment any suggestions.