Serena launched the match with three devastating aces in the first game and then kept firing them down at a demoralised Czink, who could do nothing against the mighty Williams firepower.
Shaking off her disappointing first round loss in the French Open, Serena looked razor sharp this time, moving crosscourt with effortless ease and serving a total of 10 aces.
The second set was more closely fought but one break in the ninth game was enough to secure Serena a clash in the next round with 2008 Wimbledon semi-finalist Zheng Jie of China.
Top seed Maria Sharapova overcame a second-set wobble to beat Tsvetana Pironkova 7-6 6-7 6-0 and reach the Wimbledon third round.
The match was halted on Wednesday evening with the Russian leading 3-1 in the second set but she lost the first two games to allow Bulgarian Pironkova, a Wimbledon semi-finalist two years ago, a foothold.
Pironkova relied on her powerful serve to cause Sharapova problems and the world number one double-faulted three times in the tiebreak on the way to losing it 7-3.
Sharapova, Wimbledon champion in 2004, regrouped quickly, however, and roared through the third set in 29 minutes to set up a match against Taiwan's Hsieh Su-wei.
"I started so slowly today and she came out firing today and yesterday," the French Open champion said. "It's always difficult to stop a match especially when you have the momentum.
"I wanted to start strong today but it didn't happen for me," Sharapova added. "My serve let me down in the tiebreaker."
Victoria Azarenka made no-nonsense progress into the third round with a 6-2 6-0 victory over Romina Oprandi.
Azarenka, the second seed and current Australian Open champion, needed just 51 minutes to wrap up the win.
Former Wimbledon finalist Marion Bartoli was a high-profile casualty, going down to Mirjana Lucic in straight sets.
The ninth seed had no answers to her opponent, and fell some way short of last year's effort, when she reached the quarter-finals.
14th seed Ana Ivanovic booked an intriguing third round match after she beat Kateryna Bondarenko 6-3 7-6(5). The Serb's victory means she will now face Germany's Julia Goerges in the next round after she ousted Anastasiya Yakimova in straight sets.
Angelique Kerber, the eighth seed, enjoyed a straightforward route into the last 32 with a 7-5 6-3 victory over Ekaterina Makarova on Court 18.
The German will next face 28th seed Christina McHale, who overcame Mathilde Johansson 7-5 7-5.