Most infants achieve this by 5 months. By ‘sleeping through’ this means they obtain 6 hours (or more) of sleep. Typically babies need to eat at other times during the night. There is no research to support that a baby of x weight or by 6 months should be sleeping 10 – 12 hours. The developmental goal at 5 months is 6 hours of sleep.
You likely want to get your infant to sleep through the night. Help facilitate this by bringing your baby to the breast more often (e.g. every 2 or more hours) in the early evening helps your baby store up on much needed calories so if they are ready to sleep a longer stretch they will be able to achieve this. The hind milk, which makes up more of the proportion of breast milk in the afternoon/evening, is made from rich fatty milk that has lots of calories to help sustain longer sleep for infants. Introducing formula and/or solids has not been shown in research to help with sleep.
Feed your infant at night in a dimly lit room, change his or her diaper and put them back to sleep. Help your infant sleep better by getting outside for a walk during the day and conduct daily activities in a brightly lit room. Most babies learn to decipher day from night in the first month. After this time, they can be put for their naps in a dark room during the daytime for better sleep.