My first bit of advice is to change your mindset a bit.
Off road races are not won based on how fast you can go in your highest gear. They are won based on how fast you can go through the tough stuff that requires skill and stamina. If you are going to buy a dirt bike, don't worry about its speed, worry about its suspension.
For your size and weight right now I would recommend something in the 100cc two stroke, or 125cc four stroke. These will be smaller bikes better suited for a beginner. I am concerned that if you grow another inch or two and put on a few pounds those bikes will end up being too small.
A bike that would suit you in a couple of years will be something like a Yamaha YZ-125 (two stroke) or a YZ-250F (Four stroke). You may need to lower the suspension a bit so that your feet can touch the ground.
If you think that the track will be a very rare thing then you might do well on something like a Yamaha TTR-230. This is a four stroke, has a significantly lower seat height and would be a great beginner bike. They are a much less expensive bike. You won't be impressed with its speed, however, and I don't recommend it for the track.
To get in your price range used is the only option. You should be able to pick up a 5 year old YZ-125 two stroke in your price range.
Rod