There is a James Coulter served with the Royal Artillery and a Richard Coulter served with the 1st Life Guards who are listed in the Peninsula Medal Roll 1793-1814 at The National Archivist web site at http://www.nationalarchivist.com/http://www.nationalarchivist.com/ , no Charles Coulter though.
However, there are two pensioners service records at the UK National Archives: WO 97/773/77 CHARLES COULTER Born TEMPLECARON, Donegal Served in 64th Foot Regiment Discharged aged 31 served 1810-1821
WO 97/746/128 CHARLES COULTOR alias CHARLES COULTER Born SCREEN, Sligo Served in 61st Foot Regiment Discharged aged 41 served 1826-1849
WO 97 = Royal Hospital Chelsea: Soldiers Service Documents WO 97/773 = 64th Foot: Burr - Dav ; WO 97/746 = 61st Foot: Ada - Cur
To order a copy online from red menu bar select Business Services > reprographic > order form [blue text], when you get to the actual form just enter WO97/773 then in the text box enter "copy of all pages for WO 97/773/77 CHARLES COULTER Born TEMPLECARON, Donegal" or WO 97/746 and "copy of all pages for WO 97/746/128 CHARLES COULTOR alias CHARLES COULTER Born SCREEN, Sligo" You can obtain an estimate for both at the same time [£10 deductible from order]
Alternatively you can use a researcher from red menu bar select Getting Started > Independent Researchers > choose a research area; select Army - Genealogy etc , click find researchers select from list. I've had a quick look at those with web sites and most quote an hourly rate that varies considerably from £15 to £45 per hour, you should check them yourself but the only one I have found that details the services and what you get is Jonathan Collins at http://www.militaryresearchon.com/http://www.militaryresearchon.com/ , as your supplying a NA reference to him I doubt the National Archive will beat his prices, email him to confirm. Final choice must be yours though.
Neither regiment appear to have served at Waterloo.