How's would I go about making a 70cm transmitter?
Have you considered the legal aspects? in England it would be totally illegal -even a few mW, which won't get you very far.
Semtech has lots of parts that will do that. The SX1239 SX1276 will also do 2m ham band. These are all digital modulation though -- OOK and FSK for example.
How does a RF foxhunr work???
I presume then that Adcock-style antennas as used in 'huff-duff' are employed?
Use a RFM22B which is based on a Si4432. Its an FSK data module, but put into direct mode does a reasonable FM Morse. You can set the frequency and power level between 100mw and about 2mW. There is a 1W version to. Available in 443Mhz, 868Mhz and 915Mhz versions, but can actually be programmed from 240Mhz to 930Mhz.
I think the Adcock antenna would work very well at higher frequencies - and instead of using a crt to display phase and hence direction, an I-Q synchronous demod would be the approach.