From the Mexican border near Corozal Town to Punta Gorda in Toledo, via the shortest route (Northern Highway, Boom bypass, Western Highway, Coastal Road, and Southern Highway) it is about 245 miles. All of this is paved, except the so-called Coastal Road, which is not on the coast -- if you want to avoid that, go on to Belmopan and take the Hummingbird to the Southern Highway, a little longer but a better road and far more scenic. Non-stop, averaging about 50 mph, you can make the north to south trip in about 5 hours. As a practical matter, given speed bumps, traffic, weather issues, etc. most people wouldn't want to push it that hard.
But as Diane says, very little of this route is within sight of the coast. To get to the coast you will have to take detours, and even then you are going to be within view of the water only for short distances. From memory, which may be faulty, the only places where you can see the water by main road are:
* A few blocks in Corozal Town have views of Corozal Bay
* Short distances en route to and in Sarteneja have bay views
* In Belize City there are short distances with harbor or sea views
* Going south, you can drive to the sea at Gale's Point, Dangriga and Hopkins/Sittee
* The longest stretch of sea views is down the roughly 16 miles of the Placencia peninsula, from Riversdale south, though the sea is not visible the entire way
* A short distance along PG waterfront
There are a few other places where you can briefly spot the water, such as near Monkey River, from hilltops near PG and at Barranco.
If you wanted to scope out as much of coastal areas as you could reach by car, including a side trip to Sarteneja, Gale's Point, Dangriga, Hopkins, Placencia, etc. you're talking maybe 12-14 solid hours of driving, and realistically you'd need at least two or three days, preferably longer.
Keep in mind that much of the Belize mainland coast is swamp and mangroves, not sandy beaches.