In order for a UK company to offer you a job and sponsor you for a work visa, they must demonstrate that no one else can fulfill the role from the UK and EU. So, typically, unless you have already accumulated lots of experience and/or have a rare skill and/or have a higher degree, it's rare to meet the requirements for a work visa. If you have a skill on the shortage list, that can also help, but I wouldn't base an immigration strategy around it.
ETA: Also, time spent in the UK as a student would NOT count towards ILR, so you would need the required amount of time (5 years) on a work visa, not a combination of both. And you would need ILR to apply for citizenship. (x-posted with historyenne)