The name “Tlingit” is pronounced /ˈklɪŋˌkɪt/ in English, which is approximated by the phrases “cling kit” or “clink it”. Some Tlingit people also say /ˈklɪŋˌɪt/, i.e. “cling it”, but this is less common and is not how I pronounce it.
The English name is derived from the Tlingit word Lingít (/ɬin.kít/) which means ‘person’. The unusual English spelling – “Klinkit” might be more logical – appears to have arisen as an attempt to capture the initial lateral fricative, which it should be noted is not an afficate /tɬ⁽ʰ⁾/.
The adaptation of lateral sounds like /tɬʰ/ into English as the cluster /kl/ seems to have been a regular adaptive technique in the phonology of English-speaking settlers in the Pacific Northwest during the 19th and early 20th centuries. There are numerous examples of this from Chinook Jargon placenames, such as “Kloosh Creek” from CJ /(t)ɬuʃ/ ‘good’, “Klahanie” from CJ /ɬaˈχani/ ‘outside, outdoors’, and “Klahowya Creek” from CJ /ɬaˈχawja/ ‘hello, howdy’. This only seems to have been reliable when the sound occurred in word-initial position, since word-medial instances like “potlatch” from CJ /ˈpaɬətʃ/ and “alki” from CJ /ʔaɬqi/ ‘soon’ demonstrate that other sounds would be used word-internally.
The use of /kl/ in English to approximate lateral sounds certainly explains the pronunciation of the name “Tlingit”. The spelling with initial <Tl> remains to be explained however. I feel that it is probably due to some well-intentioned recorder realizing that the lateral fricative in Lingít was not really a /kl/ sound, and hence trying to emphasise the distinctiveness of the sound with the <Tl> spelling. It may be impossible now to say why this spelling was chosen, but I feel my hypothesis presented here is probably accurate.