Although i cant cite an official view on this but i think the order of using different technologies in DRS is based on logic of eliminating the options.
DRS is mostly used in 3 situations, "caught behind wicket", "caught by close-in fielders" & "LBW".
In the first two situations above, there is no need of hawk-eye. So the sequence will/should be
1)Check for No ball
2)A slow motion close-in camera replay(If the edge is clearly visible via an actual match feed than the hotspot is just an additional closing confirmation. If not, then hotspot is the crucial confirmation)
3)Sound feed from the stump mic(Again because real feed confirmation is better than a processed technological one. If the edge is clear & distinct then hotspot will seal the deal)
4)Hotspot
Note that as yet, snickometer has not been incorporated in DRS
In case of LBW, the sequence will/should be
1)Check for No ball
2)A slow-mo close-in camera replay(To check for an inside edge before hitting the pad. If clearly visible, then the next hotspot will confirm it & hence no need of hawk-eye)
3)Hotspot(If the inside edge is visible, hotspot will be the closing confirmation. If not then hotspot evidence will be the crucial confirmation required for Not Out)
4)Hawkeye(If there is "absolutely" no clear evidence of an inside edge, Hawkeye evidence becomes crucial for decision making. In case of an "inconclusive" inside edge, a Hawkeye showing ball going wide of the stumps or not hitting in line will make hotsopt irrelevant)
In case of LBW, hawkeye is checked after hotspot because the trajectory estimate of the ball depends on where it hits first. So, in case of inside edge, it will take the hit on the bat & estimate the trajectory, which if it hits the stump, gives the impression of OUT