Hi, Nick, and welcome to the Victorian Wars Forum!
The only published history of the regiment for this period that I know of is The King's Own, The Story of a Royal Regiment, Volume Two, 1814-1914 by Colonel L I Cowper, published by the Regiment in 1939
. It's out of print, I'm afraid, but a library might be able to get it for you.
There are also the letters of Captain Jasper Hall (http://www.kingsownmuseum.plus.com/ko1157.htm
) but even the first of these is dated November 1854.
The 4th were in the Third Division, so you might have better luck following up Sir Richard England and seeing how his division were employed in that period. Some of it is easy, of course, because those dates are the same for all infantry regiments - the 14th was the landing at Eupatoria, and the march to Sevastopol began on the 19th, which was the day of the Bulganek encounter and the night before the Alma. Any deaths during this time would almost certainly be cholera rather than enemy action.
Your best hope are the regimental records, and for those you'd need to enquire at the King's Own Museum in Lancaster (http://www.kingsownmuseum.plus.com/index.htm
). If they don't have them all, they may at least have a 'Digest' you could look at.
Sorry not to be more helpful, Nick, but I'd bet the museum will have something. If you can't get to Lancaster you can try an e-mail query and see if they'll search for you for a small fee.
Best of luck!