for international to domestic connection with checked through luggage normal procedure is as follows: after arrival in BKK go to transit immigration (conjunction A/B/C/D terminal) and from there through security to the domestic terminal; in Phuket passenger will claim luggage and go there through customs (this procedure is possible for various domestic connection like Phuket, Chiangmai, Krabi, Ko Samui etc.)
for this 1 hr 15 min is sufficient; should flight arrive late or passengers miss the connecting flight, TG will book him on the next flight to Phuket (there are many flights to Phuket, every 1-3 hour)
please note that this is possible for passengers with nationality that qualify for visa exemption or valid visa
As you seem to be from India I would like to mention that citizens from India do not qualify for visa exemption. If a passenger from India do not have a valid visa, he has to do a visa on arrival in BKK and it would not be possible to get a connecting flight within 1 hr 15 min
let us know about this if you need further Information