Unless you're happy to get up REALLY early to reach the venue as it opens then I'd avoid Portsmouth.
The road from there to Goodwood is the A27 and is always full from quite early in the morning. I have sat in 4 mile traffic jams on more than 1 occasion.
There is camping on-site, but it's very expensive.
If you are happy to camp then check out the Camping & Caravanning Club's website [http://www.campingandcaravanningclub.co.uk/] for details of local campsites. If you cannot get anything immediately around Chichester then I'd look to the north around Petersfield, Midhurst & Petworth.
The alternative is to catch a train in to Chichester. There is then a bus service to the site. Return journey is £5. There is a good train service along the south coast from Southampton to Brighton, so that could potentially open up some alternative places to stay.
Paul.