This is my coin purse,'' intones actor Neil Patrick Harris, grinning like an overstocked used-car salesman, speaking in a voice as smooth as his not-yet- ready-to-shave chin. ''A very nice coin purse. It carries a lot of nice things. Look at this!'' The amateur magician opens the snap on the trick purse and reaches in. Suddenly a grapefruit-size foam ball appears in each of his hands. Then the two sponge balls become four balls. Then, just as suddenly, they all disappear.

Not a bad piece of legerdemain for a 16-year-old. But this stunt is nothing compared with the magic Harris has worked as the star of Doogie Howser, M.D. Last winter, when executive producer Steven Bochco announced he was creating a show about a teen prodigy working as a resident in a major hospital, skepticism was pandemic. But Harris's believable performance as the pubescent genius Doogie has given good health to the ABC series and boosted it to one of the highest ratings of any new show this season.

Harris doesn't claim to have the high mental voltage of his alter ego -- after all, Doogie scored perfect 800s on his SATs at 6, graduated from Princeton at 10 and from med school at 14, while Neil is only getting a 3.8 in 11th grade. But he does acknowledge that he and Howser have a few things in common. ''We're both dealing professionally with adults more than kids,'' says Harris. ''We're both involved in jobs that take up a lot of our time. And we're both 16, going through the feelings of that age. Sixteen-year-olds tend to have big mood swings that you kind of have to deal with.''

Howser and Harris have another thing in common: They both got an early start on their careers. Neil, the younger of two boys, took up acting in the fourth grade in his hometown of Ruidoso, N.Mex., 120 miles south of Albuquerque. His brother, Brian, was heading out to audition as a Munchkin in a school production of The Wizard of Oz. Harris tagged along and was tagged for Dorothy's dog, Toto, ''because they needed someone small,'' he says. Bitten hard by the bug, he began showing up wherever the greasepaint was.

''He always wanted to make people laugh,'' says Neil's father, Ron, a lawyer. Adds his mother, Sheila, also a lawyer: ''Many's the time we had company and had to say, 'Okay, Neil, that's enough.' And he'd say, 'Oh, come on, one more card trick.' ''

Harris might still be doing plays at the high school in Albuquerque, where his family moved in 1988, were it not for a stint at a summer theater camp the year before. One of the instructors was writer Mark (Children of a Lesser God ) Medoff, who had just sold a film script titled Clara's Heart. Whoopi Goldberg had signed on to play the sentimental maid, but a search was on for the spoiled boy put in Clara's charge. Medoff liked what he saw of Harris and had him do an audition tape.

Released in 1988, Clara's Heart quickly went into cardiac arrest. But it brought Harris critical respect and more jobs. Roles followed in such TV movies as Cold Sassy Tree with Faye Dunaway, Leave Her to Heaven with Patrick Duffy and a guest stint on Burt Reynolds's B.L. Stryker.

Then came the role of Doogie Howser, a part Harris couldn't call his own until a mere two days before shooting began. ''We auditioned hundreds of kids from everywhere,'' says the show's casting director, Robert Harbin. ''It was taking a big gamble no matter whom we chose, because the show wouldn't work if he didn't work in the role. When I first saw Neil, I thought, 'This is our guy,' but we still felt we had to see everybody else.''

Harris's success has obviously brought changes for his family. He and his parents have moved into a two-room apartment in Los Angeles, and Ron and Sheila have put their law careers on hold to give Neil a hand. Instead of briefs, they're helping their son write responses to the 300-400 fan letters that arrive each week.

The letters are okay by Harris. Less acceptable is the hormonal approbation accorded him by fan magazines like Super Teen, which has dubbed him the ''Hot New TV Dude'' and ''One Cute Doc.'' ''That stuff gets a little out of hand,'' says Harris, squirming as if he has just been asked to kiss a bearded lady. ''It doesn't appeal to me.''

It also makes it hard for him to feel like a regular guy -- a perfectly normal kid whose favorite activity is, say, getting together with his Albuquerque buddies and shooting video spoofs like Amish Death Spree and Amish Death Spree II, both starring Harris as a deranged Amish killer.

Besides mayhem, there's time for romance -- or at least a close friendship with Abby Wolf, 16, whom he met in an Albuquerque drama class. They're partners in an improv game called The Psychotic Boyfriend, in which he plays a nut who likes to scream and she plays the submissive girlfriend. ''We like to play the game in video stores and places like that,'' says Harris, grinning impishly. ''We're into combat theater and stuff.''

''Neil is very spontaneous,'' notes Wolf. ''It's not just some chemical imbalance in his brain.''

When he gets back to Albuquerque during the next break from the grind at Eastman Medical Center, Harris will no doubt do a bit more combat with Abby and begin work on Amish Death Spree III. ''One person's been stabbed,'' says Neil. ''Another got a five-pound weight upside the head. Someone was poisoned. Someone was choked. The hardest thing is thinking up ways to kill people off.''

So much for the Hippocratic oath.

Joanne Kaufman, Craig Tomashoff in Los Angeles, TUBE: NEIL PATRICK HARRIS FINDS A WINNING Rx IN DOOGIE HOWSER. , People, 03-19-1990