What is the weakpoint of PRNG based encryption? Let's say I used this encryption schema: 



In words instead of images:


Hash the encryption key with SHA256
Split it in half, first half use as XOR key for first 128 bit block of plaintext
Hash the second half from previous step with SHA256
Carry the hash to point 2), rinse, repeat.


What are potential problems/weakpoints of such encryption algorithm?
As a stream cipher I don't think there is any direct way to break the cipher.

First of all, it is extremely inefficient, as Richie already mentions. SHA-256 has an input block size of 512 bits, and you're only propagating 128 out of 256 output bits. So you're wasting a lot of input space. In other words, it won't be twice as slow as just SHA-256 due to the limited output being used, it will be about 8 times as slow (!).

One of the problems with your algorithm is that it only propagates 128 bits out of the internal state of the 256 bit internal state of SHA-256. That means that you are more likely to hit a cycle within the hash function. Probably your cipher mode has limited input size because of that. It is probably similar in security to CBC for a normal 128 bit block cipher in that way.

A bit of a unfair problem with it is that it has not received any formal review and that it doesn't have a formal proof. As for practical problems, a hash function input is generally not designed to take any keys, so you need to be aware of that. SHA-256 is fortunately relatively resistant against side channel attacks by itself. One good thing is that the key is not repeated anywhere in the followup hash calls.

As a cipher mode your algorithm is deterministic as it doesn't take an IV. It would however be easy to add a 128 bit IV or (rather) nonce at the start, simply as additional input next to the key. As it stands, it cannot be used as a cipher for repeated messages. You might mitigate the cycle issues by adding a counter to each invocation of SHA-256 as well, there is space enough for another 128 bit value.