The tenancy is shorthold. If it is in the fixed term you cannot evict unless the tenant breaches the terms - e g by rent being 2 months or more in arrears. If the fixed term has ended and the tenancy is periodic then you can evict by giving 2 months notice (Section 21 notice, as PP says).
Your alternative is to negotiate with the tenant, but they may well not be prepared to leave.
You cannot just put the rent up.
If you want more detailed advice contact Shelter or visit your local CAB.