It's almost impossible to die of space radiation overdose. The galactic cosmic rays can't kill you via a radiation overdose, they're dose rate is much, MUCH too low.
The only thing with a high enough dose rate is solar particle event. And, in fact, there are very few that are strong enough to kill you (but note, there are winter or thunderstorms that can easily kill you if you're unprotected on Earth). One has occurred, however, in August of 1972, with a dose of about 1 Sievert, but it'd only be that high if your only shielding was a thin space suit ( here's a source for that). If you were inside a capsule or on the surface of Mars (shielded by the yes-still-significant Martian atmosphere), you'd be totally fine. Even 1 Sv not really enough to kill you. You need about 2 Sv to really be in danger of immediate radiation overdose and death. But you could vomit in your spacesuit and suffocate. However, these events are not instantaneous, you'd have a warning and the events occur over a period of an hour or several hours, so you have enough time to get inside or behind a rock or something.
No, it's nearly impossible to die from acute natural radiation overdose in space.
You'll survive the trip. The worry is about an increase in occurrence of cancer when you get back. However, in any case, the risk of cancer from living in space is less than being a smoker. Although, given the huge deal we make about the space radiation issue, you wouldn't know it. You'd think you'd die instantly or something, which just isn't true.
As far as how to deal with it, well Mars' surface has a much lower radiation dose from GCRs and especially solar flares. You're half shielded by the planet itself and secondarily by an average of around 40 grams per square centimeter of CO2 mass, maybe more at lower altitudes. Additionally, just massive amounts of rock or dirt work great. And water is more effective per unit mass.
On the way to Mars, your best bet is to shorten the trip to 90-100 days as Musk suggests, and perhaps use your supplies (water, food, maybe propellant) to shield you from solar particle events. That'd reduce your transit dose to a manageable amount. And you can also use drugs like Amifostine to avoid some of the radiation effects, especially the effects of acute radiation (we're unsure if Amifostine helps for chronic radiation). But once on the surface of Mars, it's possible to reduce the dosage to arbitrarily low levels.
But again, these are long-term health effects, perhaps like you'd see in any kind of hazardous environment. But you'll be able to perform the mission just fine.