To complete a DNA sequence, you must complete all memories within the sequence. You can think of the Sequences as Acts and the Memories as Chapters. These are all story related and you will get them as you complete the game. You do want to worry about a different achievement called Fond Memories. This is for completing all sequences at 100% synchronization. See that achievement for more details.
Each sequence has a percentage you must fill. This achievement requires you to obtain 100% on every sequence. Each memory counts for the percentage. Some memories are simple and do not have any extra objective that can be missed. There are certain memories that will require you to do something special (an extra objective). This can vary on what is it, and when you start a memory it is detailed for you.
You can replay memories by pressing and going to DNA. As you highlight the various sequences, they will tell you what is at 100%. If you go into those sequences, you will see the memories and what objectives you need to complete. If you have trouble in any of them, you can play them later when you have more gear and skills. You can try to do them right away or you can do them later, it doesn't matter.
Follow the guide below if you need any tips or prefer to follow the list as you go along.