The ending of King Lear can be a bit tricky because there are two different versions of the play (three if you count the "conflated" text, which shmooshes the two versions together into one big, long play.) In the First Folio edition (the collected works of Shakespeare published in 1623), Lear dies thinking that Cordelia is dead and Albany gets to speak the final lines of the play. In the First Quarto edition of Lear (printed in 1608), Edgar (not Albany) delivers the final lines and Lear dies believing that Cordelia is alive.
Here's what you need to know: in all versions of the play, Lear's entire family winds up dead. What? You expected something else? This is a "Tragedy" we're talking about and all of Shakespeare's tragedies end in death. That said, Shakespeare is usually pretty good about giving the audience some glimmer of hope at the end. But, the thing about King Lear is that Shakespeare refuses to offer any kind of light at the end of the tunnel, which is why Lear is considered to be one of the most depressing tragedies ever written.
With Lear's entire family wiped out, the kingdom needs a ruler but, nobody really seems to want the job. After just about everybody dies, Albany turns to Kent and Edgar and says they must rule what's left of the kingdom together, but Kent insists that he can't, as his "master" has called him on a "journey" (5.3.11). Kent's "master" is King Lear, who is now dead, so we're left to assume that Kent plans to commit suicide. Edgar follows this up by claiming: "we that are young / shall never see so much, nor live so long" as the previous generation. In other words, the survivors of this tragedy don't seem to have much hope for the future. After the terrible loss of so many, there doesn't seem to be any way to carry on.