Q:

If Darth Vader sensed Kenobi aboard the Millennium Falcon, why didn't he go aboard? 

After a tractor beam brings the Millennium Falcon aboard the Death Star, Darth Vader senses the presence of his old teacher. 

He runs off to tell Grand Moff Tarkin that Kenobi is alive and aboard the Death Star. 

Why did he not just walk aboard the Millenium Falcon and kill everyone aboard? 

A:

Darth Vader is not in command of the Death Star; Grand Moff Tarkin is, and the decision of what to do in this kind of situation is Tarkin's not Vader's.   The appearance of Kenobi on board the station (in the very same ship that "blasted its way out of Mos Eisley"—which they may already be planning to release and track) is an important and unexpected event, which means the overall commander needs to be notified, which is just what Vader does. 
Tarkin appears to have complete confidence in Vader's ability to deal with Kenobi; Jedi matters are obvious within Vader's bailiwick.   So Vader informs his superior of what needs to done to deal with the Kenobi situation, knowing that Tarkin will accept his evaluation of what is needed.   However, had he not brought the matter to Tarkin's attention at all, that would have been going to far in exerting Vader's decision-making autonomy. 
Finally, remember that Darth Vader is confident, when he discusses the situation with Grand Moff Tarkin, that "escape is not [Kenobi's] plan."   Presumably, Vader has also inferred this information through his use of the Force.   However, if Vader knows this, it means that there is no need to hurry the confrontation with his old master.   The two will meet soon enough, when the time is right, and presumably both participants are well aware that the duel is going to happen.   Regardless of what happens to the rest of the Millennium Falcon's crew (plus the princess), Vader knows that Kenobi is not getting away. 

A:

You're forgetting the sequence of events
When Vader first arrives at the Falcon, an officer tells him straight away that the ship is empty.  Who is he going to go on board to kill? 

There's no one on board, sir.  According to the log, the crew abandoned
  ship right after takeoff.  It must be a decoy, sir.  Several of the
  escape pods have been jettisoned. 

He doesn't know Kenobi was on the ship yet, but he does strongly suspect.  However, Kenobi isn't there anymore, and the Death Star is a big place. 

I feel something, a presence I've not felt since....

Later:
                                   VADER
                               He is here...

                                 TARKIN
                     Obi-Wan Kenobi!  What makes you think 
                     so? 

                                 VADER
                     A tremor in the Force.  The last time 
                     I felt it was in the presence of my 
                     old master. 

At that same moment, the guards alert Tarkin that the princess is escaping, which confirms Vader's suspicion:

Obi-Wan IS here.  The Force is with him.  Escape is not his plan.  I must
  face him alone. 

(emphasis mine)

