Execution Flow

Identify target application: The adversary, with knowledge of vulnerable libraries or shared code modules, identifies a target application or program that makes use of these.

Find injection vector: The adversary attempts to use the API, and if they can they send a large amount of data to see if the buffer overflow attack really does work.

Techniques
Provide large input to a program or application and observe the behavior. If there is a crash, this means that a buffer overflow attack is possible.

Craft overflow content: The adversary crafts the content to be injected based on their knowledge of the vulnerability and their desired outcome. If the intent is to simply cause the software to crash, the content need only consist of an excessive quantity of random data. If the intent is to leverage the overflow for execution of arbitrary code, the adversary will craft a set of content that not only overflows the targeted buffer but does so in such a way that the overwritten return address is replaced with one of the adversaries' choosing which points to code injected by the adversary.

Techniques
Create malicious shellcode that will execute when the program execution is returned to it.
Use a NOP-sled in the overflow content to more easily "slide" into the malicious code. This is done so that the exact return address need not be correct, only in the range of all of the NOPs

Overflow the buffer: Using the API as the injection vector, the adversary injects the crafted overflow content into the buffer.

Buffer Overflow in an API Call