The NCS aims to prepare teenagers for work through team building activities and community projects.
It has run in England since 2011 and a pilot scheme began in Wales in 2014.
Mr Cameron, visiting Snowdonia, said it was a "fantastic opportunity" for young people.
The intention is that graduates on the scheme, called Teach First, lead teenagers on group activities and challenges like mountaineering, canoeing and abseiling.
More than 1,700 graduates joined Teach First in England and Wales this year, and the intention is for all to be involved in the volunteer scheme by 2017.
Visiting an outdoor education centre in the Conwy valley on Friday, Mr Cameron said: "NCS is a fantastic opportunity for young people from different backgrounds to come together to gain life skills and give something back to their local community.
"Teach First's mission to break down social barriers makes its teachers ideal volunteer mentors to engage with and inspire young people on NCS.
"This fantastic partnership will create thousands more opportunities for young people to benefit from NCS."
The NCS is a key part of Mr Cameron's "Big Society" agenda, encouraging the growth of volunteering and promoting so-called social enterprises.
The prime minister urged the Welsh government to consider taking up the scheme and offering it across Wales.
A report on the Welsh pilot - examining whether it duplicates or complements existing schemes - is expected shortly.
The UK government has been embroiled in controversy this week after one high-profile charity, Kids Company, closed its doors after ministers said they wanted to recover a Â£3m grant.
There have been suggestions that Mr Cameron himself had been "mesmerised" by the charity's founder Camila Batmanghelidjh, and had over-ruled civil service concerns over continued funding for the organisation.

A chicks' mother and father had returned to the nest in Bassenthwaite at Easter for the fourth year running, producing a trio of eggs in six days.
The first of the three chicks hatched on Thursday, with the other two birds arriving over the weekend.
But the Lake District Osprey Project said Monday was a "sad day", when the team saw only one chick was on the nest.
A spokesman said it was a "bleak" outlook for the remaining chick as it is "very likely" that the magpies would return.
It is not known if the birds, which often take scraps of fish from the osprey's nest, had eaten the chicks.
Ospreys returned to the area in 2001 after an absence of more than 150 years, and nearly 30 chicks have been raised at the nest since.
The birds fly south to Africa during late autumn, returning to the UK towards the start of April.
Graduates who do a two-year teaching scheme will be expected to volunteer with the National Citizen Service (NCS), Prime Minister David Cameron has announced.
[NEXT_CONCEPT]
Two newly hatched osprey chicks have been stolen by magpies in Cumbria.