Then casting Deputy from hand would also not trigger Tethmos correct?
Tethmos High Priest's ability won't trigger this way. Deputy of Acquittals itself doesn't target anything; with limited exceptions, no permanent spell can have targets (C.R. 115.1a-b).
In general, an ability that says "Whenever you cast a spell that targets [this permanent]" will not trigger—
if an enters-the battlefield ability (or any other ability) targets that permanent (C.R. 113.9; compare C.R. 112.1 with C.R. 113.1c; see also C.R. 603.3),
if a player other than "you" casts a spell that targets that permanent, or
if the spell cast by "you" targets only a player or only a permanent other than the one where that ability appears (C.R. 603.2, 115.1a).
Scenario: I have Tethmos High Priest and target it (doesn't really matter with what, let's say Mutagenic Growth) and return a Deputy of Acquittals to the board.
A) Can I bounce Tethmos back to my hand with Deputy?
B) Does that bounce then trigger Tethmos's effect again?
B) No. The Priest is targeted by an ability (the Deputy's trigger), not by a spell, so the heroic ability doesn't trigger.
Former Rules Advisor
"Everything's better with pirates." - Lodge
(The Gamers: Dorkness Rising)
"Any sufficiently analyzed magic is indistinguishable from science."
(Girl Genius - Fairy Tale Theater Break - Cinderella, end of volume 8)
Then casting Deputy from hand would also not trigger Tethmos correct?
In general, an ability that says "Whenever you cast a spell that targets [this permanent]" will not trigger—
See also this thread.
EDIT (Apr. 3): Edited.
EDIT (Jul. 29, 2020): Edited to conform to rule changes with Ikoria: Lair of Behemoths.