The thought of being able to abolish death as soon as you have a working BCI by running one's mind on silicon hardware is, if not ludicrous, at least really really far out. So, you hook your brain up to a computer, and get basic input and output. Great, you now have an upgraded keyboard / mouse. It doesn't get any better no matter how many capabilities you add. Direct memory access, changing the contents of programs on the fly, none of it will abolish death any more than being able to do by 'hand' does. The best you can do is create a really sophisticated program that will continue acting deterministically after the programmer goes away. You will live on in the same way Tolkien lived on after he died in his books.
You're not going to be able to abolish death until you can get neural networks to run at the speed of your current brain hardware and at similar capacity. In other words, you still have to be able to simulate a brain. To take the functions of the brain currently has, and get enough of them working electronically until you can start to build an identity on top of it.
Even once you do that, it won't feel like your brain without a lot of training, both on your artificial hardware and on your biological hardware. I envision an era where early adopters have hybrid consciousness, where we slowly incorporate an electronic identity with our biological one. Going from "This is my brain extension" to "this is another part of me, some of my thoughts are here, some of them are in this piece of hardware."
I suspect this will be a highly individual process that we will slowly gain mass competence with in a similar way we're doing with software now. We'll have to grow our ability to simulate neuronal processes and replicate our psychologies computationally. Even then, it will feel like an artificial prosthetic until you have enough capabilities to where if you suddenly lost the other side, you wouldn't feel trapped in the most hellish solitary-confinement prison ever devised.
Then after they're fully integrated, start prioritizing experiencing through the mind prosthetic. I suspect one would need many many years before cutting off the biological part wouldn't cause grievous trauma to one's sense of identity. Especially since our legal system will need a long time to figure out how to fit non-biological beings into society.