That's the perfect time to start - I did it last year. During Semana Santa, you'll need to keep an eye out for possible accommodation problems on the key days, especially in the main tourist cities. Other than that, I would not expect any problems, and you will encounter other pilgrims.
Accommodation in Seville could be very tight (and more expensive), especially on a weekend so you might want to time your start accordingly. Definitely make a reservation for Seville, and even Merida if you plan to be there during Semana Santa. (Of course, Cacares, Salamanca , Zamora, too, but you wouldn't be there yet if you start in Seville in late March.)
A few days before you get to Santiago, book a bed there. My favourite is Hospederia San Martin Pinario - you can get a "pilgrim room" at a much reduced rate if you clearly request it in an email or phone call (not using a booking service).