Construct a position which forcibly leads to White or Black being stalemated (and both events are actually possible), meaning that the position is already dead. Further (or it would be too easy) all possible lines (which you thus of course can't actually play in a game!) contain a check of White as well as of Black.
I am not sure if this is exactly what you are asking for, but here is a try.
[FEN "8/8/8/2p1p1p1/2P1PkP1/4Q1RB/3PPpPp/3n1Kbr b - - 0 1"] [startflipped ""] 1... Nxe3+
Black to move has only one legal move: 1...Nxe3 check.
Rxe3 immediately stalemates Black.
dxe3 Kxe4 3. Rf3 stalemates Black after checking Black.
dxe3 Kxg3 stalemates White after checking Black.
Since Evergalo asked, I also post a solution, where neither side is in check at the start. Problem chess etiquette says that everything that the composer does not rule out explicitly isn't cheating - for example, I used a promoted bishop, since bPh7 makes the pawn structure illegal - but ingenuity. Thus I immediately accepted the answer, especially as the "forking" into both sides' stalemate isn't as trivial as mine.
[FEN "6b1/p4p1b/4pPkr/4P2p/4Pp1P/4pR2/P3PpP1/5K1n b KQkq - 0 1"] [startflipped ""] 1... Ng3+ 2.Rxg3+ fxg3 etc.
EDIT: This is still no solution with neither side in check, since after 1...a5 the position is alive. My best try is
[FEN "Bk6/1P5p/1P6/8/7p/4p2P/4PpP1/5KBn w KQkq - 0 1"]
dead after 1.Bh2+. (Two checks still follow after that in the "proof" variants, but the position is dead only after the initial check. Kudos to the first giving a "quiet" initial position!)