What happen if I cast a targeting spell without paying the deflect cost? Will the spell still go onto the chain?

No — without paying Deflect, the play is illegal and the spell never becomes a final chain item. Deflect is a mandatory additional cost paid while you finalize the play (choosing targets and paying costs), so a spell that targets a Deflect permanent can't be legally placed on the Chain unless that cost is paid (Rule 809.1.c–d, FAQ #7667).

If you can't pay, the play is undone: the card returns to your hand and it doesn't count as "playing a spell" for triggers. If you merely forgot to pay but a legal alternative target exists, a judge may rewind only the targeting step so you choose a legal target instead of a full undo. Either way, the spell does not sit on the Chain unpaid — Deflect costs aren't applied retroactively, so once it's a finalized chain item it's too late.