Every train company has a different policy on bikes.
The London Waterloo to Portsmouth route is run by South West Trains. Their cycle policy and timetable etc is at
http://www.swtrains.co.uk/ourservices/
Generally, its difficult to take the bike on rush-hour trains. You have to book on busy routes, and it is recomended on other routes. Portsmouth out of rush hour would not be a busy route.
You dont need to pack the bike, there is either a separate guards van, or more commonly a rack for 3 bikes OR a wheelchair passenger. Note that wheelchairs always take priority, even if you were there first, so booking is recomended.
Another useful rail timetable is at:
http://www.nationalrail.co.uk/planmy..._2004128105910