The biggest problem, to my mind, is the intelligence of the thing. Being able to aggregate into a superorganism capable of intelligence... well, that's scifi, but you could see how it might be possible. Being able to disaggregate again, yep, that's fine. Being able to remain sentient, or at the very least, goal-oriented whilst disaggregated? Yeah, no.
Thought is clearly very complex. Our brains are phenomenally complex things which are formed by an extremely dense network of interconnected nerve cells. You've got about $10^{11}$ neurons in your brain, but each of those neurons is connected to an average of 7000 neighbours. Its that interconnection that enables your brain to do amazing things like discover fermentation or invent arguing on the internet. That disaggregated blob of cells has no such physical networking.
That's not to say that communication between parts of the cloud of cells is impossible. You could perhaps imagine shedding specially tagged proteins or viruses into the host body, such that only specific cells would absorb and react to their payloads. A kind of communications protocol encoded in amino acids or nucleic acids, with sender and receiver identifiers. The problem there is that the spreading of those communications packets is slow. It could take minutes or hours for the target cell to receive a specific message, and then minutes or hours again for it to send a response, even assuming the message doesn't get damaged, excreted or eaten by something else in the mean time. Compare that with nerve conduction speeds, over 100m/s. Your disaggregated swarm might be able to think, but it would do so glacially slowly.
What you need, then, is something that can form a network. My model for this would be more like a fungus or parasitic plant, capable of forming haustoria (for fungi, a specialist form of hypha, in plants a specialised root). Fungi are sessile, so clearly the sort of organism you need is something slightly different. Slime moulds are interesting, having some form of motility, but they aren't fungi. Something a little like a combination of the two might work.
Fungi can spread as spores, or perhaps by contact with an existing fungal network. They already have the means to invade host bodies, which would only be enhanced by careful immune modulation. Trying to "seep through the skin" instead of tunnelling in network threads like this is probably impractical. Motile slime-mould or amoeboid forms could get in via some other means. Naegleria fowleri gets to your brain via your nose, but the human body is well equipped with plenty of soft entrances and exposed mucous membranes... you have so many to choose from!
Infection via spores or small motile forms has a second problem with disaggregation: remembering things. A single amoeba isn't going to be able to think very hard. It does have the ability to store a good few gigabytes of information, in the form of DNA. An extra set of chromosomes or plasmids storing memories instead of being a normal part of cellular operation would clearly be required. To successfully infect a new host, many spores or infection cells would be needed to contain the whole of the organism's mind. The dumb infection vector grows and spreads and hooks up a network until it becomes big enough to think and unlock the stored information in all those DNA databases.
Evading the immune system is the second hardest issue; doing so without exposing the host to opportunistic infections and having them die on you just as you got settled in would be tricky, to say the least. Learning all of the "knowledge" encoded in the host's adaptive immune system and replicating it would be extremely non-trivial.
The best approach would be to simply not mess with that. Use an infection form capable of resisting or evading the immune system (there are lots of models for that... mycobacteria for one, HIV for another or even one of the many herpes-type viruses, but there are all sorts of intracellular parasites to take your inspiration from. Making a general purpose immune-evading strategy is probably impossible; you'd need to use your brains to tailor your attack to each new species. Just make sure that you can make your own cells (or infected host cells) produce the right kind of surface antigens that keep the host immune system happy. You could always put up with the host being super ill during the infection process... either as a result of immune reaction to your invading cells, or as a result of immune suppression allowing opportunistic infections, but just long enough for you to establish how to trick the host immune system at which point the reaction stops and they get better. Sure, weaker hosts may die, but tough hosts are the best hosts, right?
Remember, you can't make an omelette without breaking a few eggs! Maybe a whole load of humans have to die or go mad before you've perfected your technique, but you'll get there eventually. Their deaths aren't implied to kill the infecting organism too, so write up all your notes about what went well during the infection, and what didn't go so well, then disaggregate and go find some new hosts to try again on and hopefully do a better job next time.