marvel 88E8042 PCI-E Fast Ethernet Controller is not recognozed. Which ubuntu package is relevant to file a bug against?

linux-image-generic as the kernel doesn't have the driver needed, or udev isn't picking it up correctly.

According to http://cateee.net/lkddb/web-lkddb/SKY2.html there should be built-in support in the Linux kernel since version 2.6.24

What is the output of the commands

uname -a
lsb_release -crid
sudo lshw -C network
lspci -nnk | grep -iA2 net
lsmod
dmesg | grep '02:00'

HP-Compaq-6735s:~$ uname -a
Linux michael-HP-Compaq-6735s 3.19.0-22-generic #22-Ubuntu SMP Tue Jun 16 17:15:15 UTC 2015 x86_64 x86_64 x86_64 GNU/Linux
HP-Compaq-6735s:~$ lsb_release -crid
Distributor ID: Ubuntu
Description: Ubuntu 15.04
Release: 15.04
Codename: vivid
HP-Compaq-6735s:~$ sudo lshw -C network
  *-network
       description: Ethernet interface
       product: 88E8042 PCI-E Fast Ethernet Controller
       vendor: Marvell Technology Group Ltd.
       physical id: 0
       bus info: pci@0000:02:00.0
       logical name: eth0
       version: 10
       serial: 00:22:64:5b:10:2a
       capacity: 100Mbit/s
       width: 64 bits
       clock: 33MHz
       capabilities: pm msi pciexpress bus_master cap_list ethernet physical tp 10bt 10bt-fd 100bt 100bt-fd autonegotiation
       configuration: autonegotiation=on broadcast=yes driver=sky2 driverversion=1.30 latency=0 link=no multicast=yes port=twisted pair
       resources: irq:27 memory:d3100000-d3103fff ioport:3000(size=256)
  *-network
       description: Wireless interface
       product: BCM4322 802.11a/b/g/n Wireless LAN Controller
       vendor: Broadcom Corporation
       physical id: 0
       bus info: pci@0000:06:00.0
       logical name: wlan0
       version: 01
       serial: 00:21:00:5a:57:d0
       width: 64 bits
       clock: 33MHz
       capabilities: pm msi pciexpress bus_master cap_list ethernet physical wireless
       configuration: broadcast=yes driver=wl0 driverversion=6.30.223.248 (r487574) ip=192.168.178.27 latency=0 multicast=yes wireless=IEEE 802.11abg
       resources: irq:17 memory:d2000000-d2003fff
michael@michael-HP-Compaq-6735s:~$ lspci -nnk | grep -iA2 net
02:00.0 Ethernet controller [0200]: Marvell Technology Group Ltd. 88E8042 PCI-E Fast Ethernet Controller [11ab:4357] (rev 10)
 Subsystem: Hewlett-Packard Company Device [103c:30e4]
 Kernel driver in use: sky2
06:00.0 Network controller [0280]: Broadcom Corporation BCM4322 802.11a/b/g/n Wireless LAN Controller [14e4:432b] (rev 01)
 Subsystem: Hewlett-Packard Company Device [103c:1380]
 Kernel driver in use: wl
HP-Compaq-6735s:~$ lsmod
Module Size Used by
rfcomm 69632 8
bnep 20480 2
btusb 40960 0
bluetooth 491520 22 bnep,btusb,rfcomm
snd_hda_codec_hdmi 53248 1
amdkfd 81920 1
uvcvideo 90112 0
snd_hda_codec_analog 16384 1
snd_hda_codec_generic 69632 1 snd_hda_codec_analog
amd_iommu_v2 20480 1 amdkfd
radeon 1556480 3
videobuf2_vmalloc 16384 1 uvcvideo
videobuf2_memops 16384 1 videobuf2_vmalloc
wl 6369280 0
snd_hda_intel 32768 3
videobuf2_core 49152 1 uvcvideo
v4l2_common 16384 1 videobuf2_core
hid_generic 16384 0
videodev 159744 3 uvcvideo,v4l2_common,videobuf2_core
usbhid 53248 0
snd_hda_controller 32768 1 snd_hda_intel
snd_hda_codec 143360 5 snd_hda_codec_hdmi,snd_hda_codec_generic,snd_hda_intel,snd_hda_controller,snd_hda_codec_analog
media 24576 2 uvcvideo,videodev
hid 110592 2 hid_generic,usbhid
snd_hwdep 20480 1 snd_hda_codec
hp_wmi 16384 0
sparse_keymap 16384 1 hp_wmi
snd_pcm 106496 4 snd_hda_codec_hdmi,snd_hda_codec,snd_hda_intel,snd_hda_controller
snd_seq_midi 16384 0
snd_seq_midi_event 16384 1 snd_seq_midi
snd_rawmidi 32768 1 snd_seq_midi
snd_seq 69632 2 snd_seq_midi_event,snd_seq_midi
ttm 98304 1 radeon
snd_seq_device 16384 3 snd_seq,snd_rawmidi,snd_seq_midi
snd_timer 32768 2 snd_pcm,snd_seq
cfg80211 540672 1 wl
drm_kms_helper 122880 1 radeon
drm 344064 6 ttm,drm_kms_helper,radeon
kvm_amd 61440 0
snd 90112 17 snd_hwdep,snd_timer,snd_hda_codec_hdmi,snd_pcm,snd_seq,snd_rawmidi,snd_hda_codec_generic,snd_hda_codec,snd_hda_intel,snd_seq_device,snd_hda_codec_analog
kvm 483328 1 kvm_amd
hp_accel 28672 0
k10temp 16384 0
joydev 20480 0
i2c_algo_bit 16384 1 radeon
lis3lv02d 20480 1 hp_accel
serio_raw 16384 0
soundcore 16384 2 snd,snd_hda_codec
i2c_piix4 24576 0
wmi 20480 1 hp_wmi
input_polldev 16384 1 lis3lv02d
shpchp 40960 0
video 20480 0
mac_hid 16384 0
parport_pc 32768 0
ppdev 20480 0
lp 20480 0
parport 45056 3 lp,ppdev,parport_pc
autofs4 40960 2
psmouse 118784 0
sky2 61440 0
ahci 36864 2
libahci 32768 1 ahci
HP-Compaq-6735s:~$ dmesg | grep '02:00'
[ 0.832306] pci 0000:02:00.0: [11ab:4357] type 00 class 0x020000
[ 0.832374] pci 0000:02:00.0: reg 0x10: [mem 0xd3100000-0xd3103fff 64bit]
[ 0.832389] pci 0000:02:00.0: reg 0x18: [io 0x3000-0x30ff]
[ 0.832530] pci 0000:02:00.0: supports D1 D2
[ 0.832533] pci 0000:02:00.0: PME# supported from D0 D1 D2 D3hot D3cold
[ 0.832567] pci 0000:02:00.0: System wakeup disabled by ACPI
[ 3.166072] sky2 0000:02:00.0: Yukon-2 FE+ chip revision 0
[ 3.166762] sky2 0000:02:00.0 eth0: addr 00:22:64:5b:10:2a
[ 33.353502] sky2 0000:02:00.0 eth0: enabling interface
HP-Compaq-6735s:~$

Your output confirms that the Ethernet device has been recognized and a suitable driver has been loaded.

Now the question has to be: Why don't you get an IP address?

What is the output of
ifconfig -a

ifconfig -a
eth0 Link encap:Ethernet HWaddr 00:22:64:5b:10:2a
          UP BROADCAST MULTICAST MTU:1500 Metric:1
          RX packets:0 errors:0 dropped:0 overruns:0 frame:0
          TX packets:0 errors:0 dropped:0 overruns:0 carrier:0
          collisions:0 txqueuelen:1000
          RX bytes:0 (0.0 B) TX bytes:0 (0.0 B)
          Interrupt:16

lo Link encap:Local Loopback
          inet addr:127.0.0.1 Mask:255.0.0.0
          inet6 addr: ::1/128 Scope:Host
          UP LOOPBACK RUNNING MTU:65536 Metric:1
          RX packets:2128 errors:0 dropped:0 overruns:0 frame:0
          TX packets:2128 errors:0 dropped:0 overruns:0 carrier:0
          collisions:0 txqueuelen:0
          RX bytes:219494 (219.4 KB) TX bytes:219494 (219.4 KB)

wlan0 Link encap:Ethernet HWaddr 00:21:00:5a:57:d0
          inet addr:192.168.178.27 Bcast:192.168.178.255 Mask:255.255.255.0
          inet6 addr: fe80::221:ff:fe5a:57d0/64 Scope:Link
          UP BROADCAST RUNNING MULTICAST MTU:1500 Metric:1
          RX packets:3523 errors:0 dropped:0 overruns:0 frame:7696
          TX packets:3614 errors:0 dropped:0 overruns:0 carrier:0
          collisions:0 txqueuelen:1000
          RX bytes:2155187 (2.1 MB) TX bytes:675536 (675.5 KB)
          Interrupt:17

Your output shows

ifconfig -a
eth0 Link encap:Ethernet HWaddr 00:22:64:5b:10:2a
           UP

The network device is up and running.
If you plug in an Ethernet cable, do you see some indicators blinking next to the connector (usually green and/or yellow)?

The ethernet port is blinking. Today the GUI menu is working too. The network is working, but very slow. E.g. streaming Video is not working.
ifconfig -a

does not show any packet drops. I have exchanged the ethernet cables and the router port - noting seems to have a impact. My second PC on the same router, running also ubuntu, is working fine.