Q:

Didn't the USA need to declare war before launching the missiles on April 6th 2017?

The launch of over 50 missiles to strike a Syrian military base could be considered an act of war.
The UN charter states:

All members shall refrain in their international relations from the threat or use of force against the territorial integrity or political independence of any state, or in any other manner inconsistent with the purposes of the United Nations.

That should at least on paper stop member countries from engaging.
USA law has an "Authorization for the Use of Military Force" that can be issued by congress (I think?) to allow the president to take action.
Also under the War on Terror, there is an Authorization for Use of Military Force Against Terrorists that can be issued.
I haven't found mention of any of the above in media, but my English is not that good.
It states that it was a retaliation for the use of chemical weapons. Does it mean the USA and Syria are now at war?
On what grounds did the USA launch those missiles?

A:

"Could be considered an act of war."  Also could be considered other things, as well.  The unilateral nature of the action puts it on more shaky ground, but a missile strike on a base is not going to remove Assad, in and of itself, and is not in conjunction with the seizing of any territory, so the claim that it threatens the "territorial integrity or political independence" would be difficult to build a consensus around.
Within the USA, the Authorization for Use of Military Force Against Terrorists is so broad as to be a rubber-stamp for any action by the president. It was a source of concern for the opposition party and certain segments of the population from the very start, and after the end of the Bush administration, when Obama had those sweeping powers, the questioning of those broad powers gained more momentum as previous supporters suddenly found it to be troubling.
Obama cited that act as legal justification for some of his actions.  At the same time, he acknowledged that it was excessively broad, and invited Congress to visit the issue, repeal the previous one and pass a more specific, narrowly-defined Authorization related to the actions he was taking.  Congress failed to do so, for a number of reasons, and left the previous one in place.
If that were not in place, the War Powers Act, which, ironically, was meant to set boundaries on the Executive Branch use of military force, would still allow for this action, in all probability.
Even if Congress and the Courts robustly interpreted the Act (which they don't) to try and reign in presidential use of force, the letter of that law states that the president must notify Congress within 48 hours of committing armed forces to military action, and forbids the forces from remaining for more than 60 days without specific authorization for use of military force.
So, he has two days just to mention it.  Since he deployed missile strikes, it's hard to say that the forces remain committed, so he already has met the "60 day" requirement.  And, finally, under the very broad authorization against terrorism he already has the authority, with Syria being designated as an official state sponsor of terrorism.
In dealing with the UN, you will probably see the argument that, with the brazen war crime committed by Assad, there was an immediate need to deter him from further acts, and that these strikes were limited in scope so they only did that, and no more, and that the USA will defer to other bodies' deliberations, having only acted under exigent need.
NOTE: My framing of how the law is interpreted, or arguments the USA might make to support their action should not necessarily be interpreted that I am making those same arguments or claims.  Based on how similar actions have rolled out over the past 25 or so years, and discussions that arose about the authority at the time of those actions, this is how I believe the situation is, not how the world would be if I got to make all the rules.
Wikipedia: War Powers Resolution

A:

Not really an answer, but as @SDSolar notes, the Constitution states only Congress can actually declare war.
To the best of my knowledge, no court has ever decided whether the War Powers Act or any other act allowing/restricting what military actions the President can take, is Constitutional or not.
Given the number of legal challenges to President Trump's actions, it's not unforeseeable that a Congressman will bring this issue to the federal courts.
Of course, with a Republican majority in both the House and the Senate, it's quite possible a declaration of war would pass anyway, so challenging it legally might be pointless.

