My guess as to why cryonics is the best method currently available is that it's the best bet for keeping you potentially alive until better methods are developed.
Speaking just for myself, I'm not convinced that any method can give immortality (Murphy happens), but there's a good bit of hope for greatly extended lifespans.
Cryonics is useful for preserving your body until a method for immortality is developed. It is not, on it's own such a method.
If you die now, cryonics is the only method available to give you a chance at immortality.
Cryonics is either not an answer, or the only answer, depending on what exactly you mean by the question.
If you are older you should definitely be focusing on strategies for biological life extension (calorie restriction, or whatever), and everyone should sign up for cryonics as an insurance policy.
Ultimately, with full molecular nanotechnology, whether the engineering of negligible senescence is biological or digital is rather beside the point ("What exactly do you mean by ‘machine’, such that humans are not machines?" - Eliezer Yudkowsky).
However, Unfriendly AI would render the whole point moot. So the most important thing is to guarantee we get Friendly AI right.
It depends on your age. If you are under 30 it's probably staying alive long enough until we reach longevity escape velocity.. If you are over 60 it's probably cryonics.
Edited in response to Vladimir_Nesov's comment.
If you are under 30 it's almost certainly staying alive long enough until we reach longevity escape velocity.
Downvoted for the insane "almost certainly".
Directly answering the question, organ replacement including some brain augmentation that shifts into uploading eventually seems most likely. Cryonics isn't really a direct answer to the question, if you want to talk about trajectories to achieve immortality.... It's too hard to predict where a breakthrough will be made or a wall will be hit. I think the most feasible trajectory is focusing on money and power, then organ replacement and traditional life extension techniques, which include general existential risk reduction.
"Mind uploading" makes debatable assumptions about how the mind works. It might have the result of killing you while leaving behind a Siri-like app which tricks living people's theory of mind into thinking that you have survived the upload.
Cryonics, by contrast, falls into the realm of testable neuroscience, as Sebastian Seung argues in his new book:
I don't think the cryonics of today are a very good bet for immortality, and mind-uploading may still take a while to develop... So I guess your best bet is to use the currently known methods to improve your life expectancy, and hope for the situation to improve during your lifetime (or even better, work on it!).
I'm not sure what criteria you're intending with "feasible", but I'd say FAI, as uploading/cryonics have a lot of failure modes, one of which is uFAI. Unless something weird happens, e.g. a currently-hidden AI keeps us from gobbling the stars, then an FAI once unleashed should be able to revive every human who's ever died, so even if you die before it's developed you should still be okay. (If an FAI would want to do that, anyway.) Whereas most people would be skeptical that an AI could be powerful enough to resurrect every human ever, I'm actuall...
What looks, at the moment, as the most feasible technology that can grant us immortality (e.g., mind uploading, cryonics)?
I posed this question to a fellow transhumanist and he argued that cryonics is the answer, but I failed to grasp his explanation. Besides, I am still struggling to learn the basics of science and transhumanism, so it would be great if you could shed some light on my question.