I have done a lot of solo PVE in lowsec. In my opinion you have a couple options. The best option is covert ops cloaking and using a ship like the Stratios or Tengu. If these are not an option, but you do have access to normal cloaking devices, you can bring most ships to lowsec using the MWD+Cloaking trick to get through gates. This would be ideal because you can bring larger PVE ships which will allow you to kill the tougher clone soldiers in asteroid belts and complete the higher level combat sites (cosmic signatures, escalations, etc.)
If you don’t have access to cloaking, you can use a ship that aligns to warp in less than two seconds. This makes you nearly impossible to catch and you can warp from system to system going through gates without really worrying about anything.
Whichever method of travel you use, find a system deep in lowsec with nobody in local. Try belt ratting, do all of the anomalies, etc. Occasionally you can do so in more populated systems. Keep practicing evasive tactics as suggested by others in this thread.
Here is an example of a ship you can use for lowsec PVE with the MWD+Cloak trick. You need to put the MWD on it when travelling and probably turn off the hardener to save CPU.
[Vexor, LowSec PVE]
Damage Control II
Drone Damage Amplifier II
Drone Damage Amplifier II
Drone Damage Amplifier II
Co-Processor II
Limited Adaptive Invulnerability Field I
Large Shield Extender II
Medium Ancillary Shield Booster
10MN Monopropellant Enduring Afterburner
Dual 150mm Prototype Gauss Gun
Dual 150mm Prototype Gauss Gun
Dual 150mm Prototype Gauss Gun
Improved Cloaking Device II
Medium Anti-EM Screen Reinforcer I
Medium Anti-EM Screen Reinforcer I
Medium Anti-Thermal Screen Reinforcer I
Hobgoblin II x5
Federation Navy Hammerhead x5
Caldari Navy Vespa x5
Cap Booster 50 x107
Iridium Charge M x2480
Lead Charge M x2000
Antimatter Charge M x2000
50MN Cold-Gas Enduring Microwarpdrive x1
There are a couple ships capable of aligning to warp in less than 2 seconds that are available to alpha clones. I used to use the Daredevil with polarized blasters which I don’t recommend unless you are just bored and looking to do something ridiculous. The one ship I have settled on for this task is the Sunesis. Below is an example Sunesis fitting for solo lowsec PVE. You have to remove the combat modules in the low slots and replace them with inertial stabilizers while travelling. Then, with the right skills, the ship will align in 1.9 seconds.
[Sunesis, Sunesis LowSec PVE]
IFFA Compact Damage Control
Vortex Compact Magnetic Field Stabilizer
Vortex Compact Magnetic Field Stabilizer
Co-Processor II
Limited Adaptive Invulnerability Field I
Medium Azeotropic Restrained Shield Extender
Medium Ancillary Shield Booster
1MN Monopropellant Enduring Afterburner
Light Ion Blaster II
Light Ion Blaster II
Light Ion Blaster II
Rocket Launcher II
Small Anti-Kinetic Screen Reinforcer I
Small Anti-Kinetic Screen Reinforcer I
Small Anti-Thermal Screen Reinforcer I
Hobgoblin II x8
Cap Booster 50 x70
Void S x3361
Null S x3000
Scourge Rage Rocket x2050
Inertial Stabilizers II x3