In latin, ch was used to transcribe the Greek Χ (Chi) when it had a /kʰ/ sound (some regional forms of the alphabet used that letter where Ξ was used elsehwere, which is why our X looks like Χ).
This has left its mark on some English word, notably Christ and words derived from it (Christian, Christmas).
English pulchritude from Latin pulchritudo from Latin pulcher (beautiful) + -tūdō (makes an abstract noun from an adjective like English -ity) is straightforward.
Where pulcher came from is unknown, and hence is why it had a ch. Maybe it was from a now lost Greek word. Maybe it had a /k/ sound or some other sound that got mutated and the spelling changed to match the /kʰ/ sound. Maybe it was borrowed from someone else entirely. Maybe some Roman bloke came up with some theory about why pulcher was more logical ;)
In any case, it did have the /kʰ/ sound, just as the English now has a /k/ sound of those ch words we got from Greek via Latin (again, Christ being another example).