Trapper shows Hawkeye a photo of his daughters, but the photo is in 1970s-quality color, with the girls dressed in '70s styles and striking a "K-Mart Special" pose that was not used for portraits in the 1950s.
After Klinger reads his "letter from home" to Col. Blake, Henry reaches around to get Klinger's file. When he turns back, he starts to open the folder. The camera angle then changes to over Henry's shoulder, and we see the folder is still closed and he proceeds to open it a second time.
When Klinger talks to Henry in his office with his "letter from home", Henry reaches around and gets a folder with his previous "letters from home". But the "letters" Henry takes from the folder are not creased, so even if Klinger had faked them, they certainly would've been creased from being in an envelope, which the one that Klinger read was.