No, a president using military force isn't illegal notwithstanding the Constitution granting Congress the power to declare war. The legality is clearly more complicated given that the executive waged war on American Indian tribes through the 19th century without them and given that early in our history, in the Quasi-War with France and the Tripolitan War, we waged war without a declaration of war. But it is an interesting question.
Congressional approval of war is a more nuanced standard than you may want to believe. As the commander-in-chief, the president has the power to use the military.
Sure, Congress has the power to declare war--an affirmative approval--but as the history shows, our military has been sent into action without that document since our founding as a nation. You'd think the founders would have been more outraged at that, no?
By funding military operations or failing to ban specific actions, Congress passively approves president-initiated military action.
And while getting Congressional active approval is nice to rally support for the war effort, even that isn't a clear advantage when you consider that Democrats who supported the Iraq War turned on a dime to oppose it (and deny they supported it and the valid and lawful reasons for war) when the going got tough.
Also, declaring war was once part of a large body of law on waging war, so declaring war was a necessary thing to wage war.
Since the UN Charter, that is rather pointless since war is in theory outlawed, only possible as a defensive measure or as authorized by the UN (I believe Korea--and that is an open war merely suspended by an armistice please recall for your pucker factor enjoyment in the current Korean crisis--and the Persian Gulf War are the only two examples of that) to repel an unlawful attack.
(I've also read on Strategypage that a formal declaration of war by Congress triggers all sorts of domestic laws designed to recreate World War II powers without counting on Congress that might be nuked to stick around long enough to introduce and pass all of them--just declare war and scatter! But I've never found anything else backing that and even searched online federal statutes without luck. So I don't know what to make of that claim.)
But consider one factor that I've never seen raised. The creation of a large standing military has changed the assumptions that existed in the 18th century when our fuzzy war powers were written into our Constitution.
At our nation's birth, the president had the power to use the small American military as he saw fit. And whether it was the naval services in the Quasi-War and Tripolitan War or ground forces in Indian wars, presidents did just that without constitutional crises.
And if America was invaded, the president needed to be able to react fast without waiting for Congress to convene and approve fighting. Besides, the state militias would be called up by the states as the first line of defense given the small size of the American Army.
But what about a foreign war? The Navy was small. Militias couldn't be sent abroad. And the Army and Marine Corps were small. Despite the theoretical power of the president to start a war, under the real world circumstances of the time, the practical limitation was that Congress would have to authorize the creation of an army and navy to wage a foreign war. So a Congressional affirmative consent was needed.
That distinction was probably well understood at the time and accepted, clearly.
But now the distinction between what is allowed in theory and what is practically possible no longer exists to a large degree. Since World War II, the president has a large and capable military under command that can use that fuzzy authority to wage war absent Congressional action to stop the president.
I'm no lawyer. I'm a history and political science major. And I had but one international law class that I've largely forgotten.
But the match between Constitutional powers and military capabilities that existed for most of our history and the divergence since World War II seems to be the biggest problem in the whole "what is legal" question we debate these days over war powers, and which largely seems to follow what party you are for and what party you are against (not me--I never argued that the Libya War violated our Constitution as many Republicans did in 2011).
And again, note that we are already at war with North Korea under the blessings of the United Nations that has repeatedly sanctioned North Korea for missile and nuclear programs.