International students on Tier 4 visas do not have an automatic right to remain and work in the UK after their studies.
The requirements are strict and options are limited, so you should research the different options as early as possible so that you are prepared. You will need to qualify for one of the working visa schemes offered by UK Visas and Immigration.
Most graduates will need to investigate Tier 2 visas, which require them to already have a graduate job offer from a UKVI-registered sponsor, but PhD students should also check out the Doctorate Extension Scheme, which is a good entry point for an early career researcher and more flexible. There are also options that apply specifically to particular nationality groups or for internships.
The working after your studies section of our website will give you a bit more information about the various options, but you will then need to research your chosen options further using the detailed resources on the UKVI website.
The immigration advisers in the International Student Support Office specialise in student rather than working visas, but they can give you some initial advice on this topic. Make sure you read the above resources before you log an enquiry.