Seras' character goes through one of the most formidable development arcs in the series. It's big, it's dramatic, and we get a front-row seat for all the major events.
For starters: Like everyone in Hellsing, Seras had a traumatic childhood. A Zorin-induced flashback shows one night in which her mother told her to hide in a closet as the house was robbed; from here Seras watched as her parents were shot by the burglars. She made them regret it - with some creative use of a fork - but she was left an orphan (again, like everyone else in Hellsing).
The young Seras didn't do well at the orphanage. She didn't get along with the other kids; she tended to beat up the boys. She ended up a police officer at nineteen, one of those deceptively mild people whose turbulence is hidden under the surface. Her comrades nicknamed her "Kitten".
In the TV anime, Seras is a member of special unit D-11; in the manga and OVA, she's just a generic police officer. In every version, she's young and scared when the mission to the village of Cheddar finds ghouls walking around. Of course, she's also the only officer who doesn't get turned into one.
Alucard, finally sent to Cheddar to deal with the local vampire properly, appreciates this. When said local vampire captures Seras and uses her as a shield, he doesn't hesitate to shoot through her - but he then asks if she'd like to come with him.
Seras doesn't have a lot of time to make this decision. She says yes.
She'll wrestle a lot with this hasty decision afterwards. Vampires are well-known in pop culture, but Seras didn't realize all the implications of actually becoming one. She refuses to drink blood, even the medical blood supplied to Hellsing for the purpose. She can't turn into a bat or walk through walls; her power seems limited to inhuman strength and speed, and she's reluctant to use even that.
Nevertheless, Alucard tells Walter that she'll drink eventually - and that she has all the makings of a great undead. (Until then, he refuses to call her anything but "police girl"). The Major tells his soldiers that Seras has no idea of her power, and that he considers her as formidable an opponent as Alucard and Integral.
In the TV anime, Seras loses control and drinks a few times, regretting it afterwards. She can't eat, but she tries. In the manga, Walter notices this and tells Integral, who corners Seras in the dining room and proceeds to cut her own finger. Seras ends up strengthened by a nourishing snack of 100% virgin blood.
The TV anime leaves Seras' situation unresolved, although it introduces the element of Helena. The only vampire Seras has met besides her intimidating master who's worth talking to, Helena's a good choice for someone to talk to. Her composure and wisdom make her respectable; her child's body brings home her immortality. But Helena's soul is rent from her body before she and Seras can sit down for a heart-to-heart chat.
The manga introduces the complication of Pip Bernadotte, who brings a new lightness and cheerfulness to her life. Between his good-natured teasing and his surprisingly deep sense of honor, he irritates her at first, but she ends up in love with him. And then he dies.
Pip gets Seras to do what Alucard had been encouraging her to do all along: become True Undead. Scythed by Zorin Blitz, he lies dying in her arms and urges her to drink his blood. She does. Draining Pip of blood, she also absorbs his soul, as Alucard has done with armies of people over the ages; this confuses Zorin, who sees into souls in order to send people into flashbacks. And then Zorin gets torn apart by Seras/Pip, and it doesn't matter much any more.
There's a new True Undead flying around London. Her eyes are red and her uniform scarlet with blood. She's no longer even imitating a human. One arm, lost to Zorin's scythe, has been replaced by a night-black tendril of shadow that occasionally takes the form of a batlike wing. She soars across the sky at will, and bares her teeth at anyone who would oppose Hellsing.
When Alucard sees her like this, he finally calls her by her name: Seras Victoria.
(And then he pats her on the head, and she smiles a cute smile. Hellcat or not, she's still Seras.)