A christening is when a child (usually) is welcomed into the family of a Christian church. (If it's an adult we'd usually use the word baptism instead)
The Church of England is set up so that everyone has a local parish church so use the website www.achurchnearyou.com/parishmap.php to find yours. Whether you can have a christening for a baby without the parents having any faith themselves depends on the minister in charge of that church. Some take the attitude that just wanting your baby christened is a welcome start towards thinking about your own relationship with God and will be happy to agree (though they may want you to come to church and/or some classes a few times to find out what the words you will say actually mean). Other ministers will prefer to guide you towards holding a ceremony of thanksgiving, naming and welcome which isn't a christening.
The christening ceremony itself requires you and the godparents (who must themselves be Christian usually) to affirm for yourselves and on behalf of the child that you believe and trust in God as Father, Son and Holy Spirit, and that you turn away from evil and towards Christ and promise that you will bring up the child as a Christian. Whether your own beliefs fit with that will be a very personal decision which the minister will help you with.
(Oh and there's not usually any fee if the christening takes place during a normal service. There probably would be if you are having a special ceremony which is just for you and your family. Even if they don't have a fee, or if they keep the fee as low as possible to ensure it is affordable, they will probably hope (and possibly explicitly encourage) that you make a donation towards the church in some way. However happy they are to do this for you, churches do cost hundreds of thousands of pounds a year to run and maintain and get no government funding to do so despite there being plenty of government legislation fixing what they can and can't do so as to keep those costs high.)
There should be no fee at all for an Anglican Christening, whether private or public - there is only a £13 fee if you want a certificate of Baptism according to the current Table of Fees. The Church properly takes the view that there should be no financial reason for a person, whether adult or child, not to become a member of it. Obviously any donation would be welcomed by any Church as itlypocerka says.
In fact anybody can baptise a child - you do not need to be in Holy Orders. It is the one Sacrament that can be performed by a layman - no doubt seen as a necessity in days when baptism was seen as removing original sin and as a prerequisite to Salvation and many babies died in the minutes after birth.
Christening is explicitly a christian rite, though as PP have explained, the vicar in your parish may be perfectly happy if you just want to do it as a traditional ceremony, and you and the godparents don't have any issue with the promises.